Small Indentations of Rubber Blocks: Effect of Size and Shape of Block and of Lateral Compression

2006 ◽  
Vol 79 (4) ◽  
pp. 674-693 ◽  
Author(s):  
A. N. Gent ◽  
O. H. Yeoh
Keyword(s):  
Cross Section ◽  
Strain Energy ◽  
Rectangular Cross Section ◽  
Strain Energy Function ◽  
Elastic Solid ◽  
Biaxial Compression ◽  
Rigid Surface ◽  
Lateral Compression ◽  
Block Width ◽  
Rubber Block

Abstract Many gaskets and seals consist of a long rubber strip or thin-walled ring, placed on a flat rigid surface and indented by a flat-ended rigid indenter. We have examined their resistance to small indentations by FEA. They are treated as infinitely-long elastic blocks of rectangular cross-section, resting on a rigid frictionless base. The indentation stiffness is calculated for various ratios of indenter tip width to block width and to block thickness, using two restraint conditions on the outer surfaces: frictionless walls (zero outwards displacement), as for a gasket placed in a recess; or stress-free, as for a gasket with no lateral restraint. For an infinitely-wide and infinitely-thick block, the theoretical resistance to indentation is zero. For comparison, the indentation stiffness is calculated for cylindrical rubber blocks of varied radius and thickness, indented by a flat-ended cylindrical indenter. In this case the result for an infinitely-large block is finite. A second study treats indentation of a rubber block, pre-compressed in the surface plane. Biot showed that the indentation stiffness of a half-space becomes zero at a critical compression, about 33% for equi-biaxial compression and 44 % for plane strain compression, for both a neo-Hookean and a Mooney-Rivlin elastic solid. FEA calculations were made of the indentation stiffness of neo-Hookean blocks of various sizes, pre-compressed to various degrees. The results are compared with Biot's result. In an Appendix, the critical degree of compression is calculated for a more realistic strain energy function than either the neo-Hookean or the Mooney-Rivlin approximation.

scholarly journals On an approximate solution for the bending of a beam of rectangular cross-section under any system of load, with special reference to points of concentrated or discontinuous loading

1902 ◽  
Vol 70 (459-466) ◽  
pp. 491-496
Keyword(s):  
Approximate Solution ◽  
Special Reference ◽  
Cross Section ◽  
Rectangular Cross Section ◽  
Elastic Equilibrium ◽  
Elastic Solid ◽  
Two Dimensions ◽  
Two Dimensional ◽  
Applied Stresses

The paper investigates the elastic equilibrium of a long bar of rectangular cross-section in those cases where the problem may be treated as one of two dimensions, namely:— ( a .) When the strain being in the plane of xy , the elastic solid extends indefinitely in the direction of the applied stresses over the bounding planes y = ± b , x = ± a being the same for any two sections parallel to the plane of xy . We then have a strictly two-dimensional strain.

The finite compression of elastic solid cylinders in the presence of gravity

1971 ◽  
Vol 321 (1546) ◽  
pp. 381-396 ◽  
Keyword(s):  
Axial Compression ◽  
Energy Function ◽  
Strain Energy ◽  
Body Force ◽  
Strain Energy Function ◽  
Elastic Solid ◽  
Infinitesimal Deformation ◽  
Material Constants ◽  
Gravity Effects ◽  
Theoretical Predictions

The deformation induced by gravity in a solid cylinder is considered. The cylinder is placed on a horizontal frictionless surface with the axis vertical and is treated as an isotropic incompressible elastic material. Further distortion is produced by finite axial compression. Thus the overall deformation is predominantly uniform with a small perturbation superimposed to account for the gravity effects. A particular case is when there is no finite axial compression. The solution then describes the shortening and general infinitesimal deformation associated with the gravitational body force. The results may be used for determining the material constants of soft elastic materials. In particular, the equivalent of Young’s modulus for gelatin has been found by measuring the changes in height of cylindrical specimens when removed from rigid containers and using the appropriate formula derived in the analysis. The results obtained were extremely consistent. In addition, the behaviour of gelatin under finite compression has been examined. By comparing the theoretical predictions with the experimental measurements it is shown that gelatin behaves more as a Mooney material than as a material which has a quadratic form for the associated strain energy function. Values of the two material constants occurring in the Mooney form of the strain energy function are obtained.

Deflection of a Thick Ring in Diametral Compression by Test and by Strength-of-Materials Theory

1959 ◽  
Vol 26 (2) ◽  
pp. 294-295
Author(s):  
Alexander Blake
Keyword(s):  
Cross Section ◽  
Strain Energy ◽  
Rectangular Cross Section ◽  
Theoretical Estimate ◽  
Shear Stresses ◽  
Strength Of Materials ◽  
Diametral Compression ◽  
Normal And Shear Stresses

Abstract Experimental and theoretical deflection studies are briefly described for several steel rings, of uniform rectangular cross section, compressed by two forces along a diameter and having D/d ratios ranging from 1.3 to 1.9. The calculations are based on the principle of Castigliano and expressions for strain energy due to bending, normal, and shear stresses. Discrepancies between the theoretical estimate and the tests are shown.

On the use of convected coordinate systems in the mechanics of continuous media

1951 ◽  
Vol 47 (3) ◽  
pp. 575-584 ◽  
Author(s):  
A. S. Lodge
Keyword(s):  
Coordinate System ◽  
Strain Energy ◽  
Equations Of State ◽  
Equations Of Motion ◽  
Strain Energy Function ◽  
Elastic Solid ◽  
Moving Medium ◽  
Coordinate Systems ◽  
Invariance Properties ◽  
The Right

The use of a coordinate system convected with the moving medium for describing its mechanics, first proposed by Hencky (5), has since been extended by several authors, and has several advantages over the more conventional use of a coordinate system fixed in space; Brillouin(1) has shown that the relation between the strain-energy function for an ideally elastic solid and the stress tensor takes a very simple form when the latter is referred to a convected coordinate system; Oldroyd(8) has given a very general discussion of the formulation of rheological equations of state and has shown that the right invariance properties are most readily recognized when the equations are referred to a convected coordinate system; Green and Zerna (4) have similarly expressed the equations of motion and boundary conditions; and Gleyzal (2), and Green and Shield (3) have applied the formalism to certain problems in elasticity theory.

scholarly journals Antiplane wave scattering from a cylindrical cavity in pre-stressed nonlinear elastic media

2015 ◽  
Vol 471 (2182) ◽  
pp. 20150450 ◽  
Author(s):  
Tom Shearer ◽  
William J. Parnell ◽  
I. David Abrahams
Keyword(s):  
Cross Section ◽  
Cylindrical Cavity ◽  
Strain Energy ◽  
Wave Equations ◽  
Strain Energy Function ◽  
Scattering Cross Section ◽  
Radial Deformation ◽  
Scattering Cross ◽  
Scattered Power ◽  
Longitudinal Stretch

The effect of a longitudinal stretch and a pressure-induced inhomogeneous radial deformation on the scattering of antiplane elastic waves from a cylindrical cavity is determined. Three popular nonlinear strain energy functions are considered: the neo-Hookean, the Mooney–Rivlin and a two-term Arruda–Boyce model. A new method is developed to analyse and solve the governing wave equations. It exploits their properties to determine an asymptotic solution in the far-field, which is then used to derive a boundary condition to numerically evaluate the equations local to the cavity. This method could be applied to any linear ordinary differential equation whose inhomogeneous coefficients tend to a constant as its independent variable tends to infinity. The effect of the pre-stress is evaluated by considering the scattering cross section. A longitudinal stretch is found to decrease the scattered power emanating from the cavity, whereas a compression increases it. The effect of the pressure difference depends on the strain energy function employed. For a Mooney–Rivlin material, a cavity inflation increases the scattered power and a deflation decreases it; for a neo-Hookean material, the scattering cross section is unaffected by the radial deformation; and for a two-term Arruda–Boyce material, both inflation and deflation are found to decrease the scattered power.

Finite extension and torsion of cylinders

1951 ◽  
Vol 244 (876) ◽  
pp. 47-86 ◽  
Keyword(s):  
Hydrostatic Pressure ◽  
Cross Section ◽  
Cross Sections ◽  
Complex Variable ◽  
Energy Function ◽  
Strain Energy ◽  
Strain Energy Function ◽  
Finite Extension ◽  
Order Effects

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.

Surface Waves Guided by the Exterior of a Rectangular Elastic Solid

1986 ◽  
Vol 53 (2) ◽  
pp. 379-381 ◽  
Author(s):  
A. K. Gautesen
Keyword(s):  
Surface Waves ◽  
Cross Section ◽  
Cross Sections ◽  
Rectangular Cross Section ◽  
Elastic Solid ◽  
Dispersion Relations ◽  
Ray Theory ◽  
Elastic Bar

We show that surface waves can be guided on the exterior of an isotropic elastic bar with a rectangular cross section. We assume that the dimensionless wavenumber is sufficiently large that elastodynamic ray theory is valid. Dispersion relations are obtained and representative curves for various cross sections are shown.

A Preliminary Analysis of the Data From an in Vitro Inflation-Extension Test Can Validate the Assumption of Arterial Tissue Elasticity

2013 ◽  
Vol 135 (8) ◽  
Author(s):  
Alexander Rachev ◽  
Tarek Shazly
Keyword(s):  
Experimental Data ◽  
Renal Artery ◽  
Strain Energy ◽  
Mechanical Response ◽  
Strain Energy Function ◽  
Elastic Solid ◽  
Tissue Elasticity ◽  
Biaxial Stretching ◽  
Constitutive Formulation

The objective of this study is to propose a method for preliminary processing of the experimental data from an inflation-extension test on tubular arterial specimens. The method is based on the condition for existence of a strain energy function (SEF) and can be used to verify whether the data from a certain experiment validate the assumption that the tissue can be considered as an elastic solid. As an illustrative example of the proposed method, experimental data for a porcine renal artery are used and the sources of the error in satisfying the condition of elasticity are analyzed. The results lead to the conclusion that the experimental data for a renal artery validate that the artery exhibits an elastic mechanical response and a constitutive formulation based on the existence of the SEF is justified. A modification of the proposed method for the case of an in-plane biaxial stretching test of mechanically isotropic and orthotropic tissues is considered.

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