Bifurcation of rotating circular cylindrical elastic membranes

1980 ◽  
Vol 87 (2) ◽  
pp. 357-376 ◽  
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.

Analytical and Experimental Study of Beam Torsional Stiffness With Large Axial Elongation

1988 ◽  
Vol 55 (1) ◽  
pp. 171-178 ◽  
Author(s):  
M. Degener ◽  
D. H. Hodges ◽  
D. Petersen
Keyword(s):  
Experimental Data ◽  
Axial Force ◽  
Energy Function ◽  
Strain Energy ◽  
Strain Energy Function ◽  
Torsional Stiffness ◽  
Deformation Tensor ◽  
Axial Elongation ◽  
Green Strain ◽  
Strain Components

The axial force and effective torsional stiffness versus axial elongation are investigated analytically and experimentally for a beam of circular cross section and made of an incompressible material that can sustain large elastic deformation. An approach based on a strain energy function identical to that used in linear elasticity, except with its strain components replaced by those of some finite-deformation tensor, would be expected to provide only limited predictive capability for this large-strain problem. Indeed, such an approach based on Green strain components (commonly referred to as the geometrically nonlinear theory of elasticity) incorrectly predicts a change in volume and predicts the wrong trend regarding the experimentally determined axial force and effective torsional stiffness. On the other hand, use of the same strain energy function, only with the Hencky logarithmic strain components, correctly predicts constant volume and provides excellent agreement with experimental data for lateral contraction, tensile force, and torsional stiffness—even when the axial elongation is large. For strain measures other than Hencky, the strain energy function must be modified to consistently account for large strains. For comparison, theoretical curves derived from a modified Green strain energy function are added. This approach provides results identical to those of the Neo-Hookean formulation for incompressible materials yielding fair agreement with the experimental results for coupled tension and torsion. An alternative approach, proposed in the present paper and based on a modified Almansi strain energy function, provides very good agreement with experimental data and is somewhat easier to manage than the Hencky strain energy approach.

Problems in determining the elastic strain energy function for rubber

2010 ◽  
Vol 45 (3) ◽  
pp. 232-235 ◽  
Author(s):  
C. Nah ◽  
G.-B. Lee ◽  
J.Y. Lim ◽  
Y.H. Kim ◽  
R. SenGupta ◽  
...  
Keyword(s):  
Energy Function ◽  
Elastic Strain ◽  
Strain Energy ◽  
Elastic Strain Energy ◽  
Strain Energy Function

A Mathematical Model of Lung Parenchyma

1980 ◽  
Vol 102 (2) ◽  
pp. 124-136 ◽  
Author(s):  
A. D. Karakaplan ◽  
M. P. Bieniek ◽  
R. Skalak
Keyword(s):  
Strain Energy ◽  
Strain Energy Function ◽  
Strain Curve ◽  
Lung Parenchyma ◽  
The Finite Element Method ◽  
Stress Strain Curve ◽  
Proposed Model ◽  
Elastic Membranes ◽  
Mammalian Lung ◽  
Nonlinear Elastic

The geometry of the proposed model of the parenchyma of a mammalian lung reproduces a cluster of alveoli arranged around a lowest-level air duct. The alveolar walls are assumed to be nonlinear elastic membranes, whose properties are described in terms of a strain energy function which reflects the hardening character of the stress-strain curve. The effect of the surfactant is included in terms of a variable (area-dependent) surface tension. Analyses of various mechanical processes in the parenchyma are performed with the aid of the finite element method, with the geometric and physical nonlinearities of the problem taken into account.

Large deformation possible in every isotropic elastic membrane

1977 ◽  
Vol 287 (1341) ◽  
pp. 145-187 ◽  
Keyword(s):  
Strain Energy ◽  
Closed Surface ◽  
Elastic Membrane ◽  
Energy Response ◽  
Normal Surface ◽  
Metric Tensor ◽  
Alternative Description ◽  
Special Cases ◽  
Static Solutions ◽  
Elastic Membranes

This paper is concerned with static solutions of finitely deformed elastic membranes regarded as thin shells. It deals with deformations that can be maintained, in the absence of body force, in every isotropic elastic membrane by the application of edge loads and/or uniform normal surface loads on the major surfaces of the thin shell-like body. The solutions, which are valid for both compressible and incompressible materials, are obtained with the use of a strain energy response function which depends on the metric tensor of the membrane in its deformed configuration. The main results are summarized by several theorems and their corollaries in accordance with three mutually exclusive cases for which the initial undeformed surface of the membrane (which may be a sector of a complete or closed surface) is, respectively, developable, spherical and a surface of variable Gaussian curvature satisfying certain differential criteria. The corresponding deformed surfaces are, respectively, a plane or a right circular cylinder, a sphere and a surface of constant mean curvature. These results are exhaustive in that they represent all finite deformation solutions possible in every isotropic elastic material characterized by the strain energy response mentioned above. Also discussed are some special cases of the general results and several families of solutions in terms of an alternative description which should be useful in application and which permit easy interpretations.

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

1993 ◽  
Vol 34 (3) ◽  
pp. 296-317 ◽  
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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