An Integral Bound on the Strain Energy for the Traction Problem in Nonlinear Elasticity with Small Strains.

1980 ◽  
Author(s):  
Joseph J. Roseman
Keyword(s):  
Nonlinear Elasticity ◽  
Strain Energy ◽  
Integral Bound ◽  
Small Strains

An integral bound for the strain energy in nonlinear elasticity in terms of the boundary displacements

1979 ◽  
Vol 9 (1) ◽  
pp. 21-27 ◽  
Author(s):  
Shlomo Breuer ◽  
Joseph J. Roseman
Keyword(s):  
Nonlinear Elasticity ◽  
Strain Energy ◽  
Integral Bound

Integral estimates for the displacement and strain energy in nonlinear elasticity in terms of the body force

1977 ◽  
Vol 15 (5) ◽  
pp. 317-322 ◽  
Author(s):  
Marian Aron ◽  
Joseph J. Roseman
Keyword(s):  
Nonlinear Elasticity ◽  
Strain Energy ◽  
Body Force ◽  
The Body

scholarly journals Methodical fitting for mathematical models of rubber-like materials

2017 ◽  
Vol 473 (2198) ◽  
pp. 20160811 ◽  
Author(s):  
Michel Destrade ◽  
Giuseppe Saccomandi ◽  
Ivonne Sgura
Keyword(s):  
Experimental Data ◽  
Mathematical Models ◽  
Nonlinear Elasticity ◽  
Strain Energy ◽  
Nonlinear Response ◽  
Energy Functions ◽  
Nonlinear Elasticity Theory ◽  
Polymeric Chains ◽  
Strain Energy Functions ◽  
Global Fitting

A great variety of models can describe the nonlinear response of rubber to uniaxial tension. Yet an in-depth understanding of the successive stages of large extension is still lacking. We show that the response can be broken down in three steps, which we delineate by relying on a simple formatting of the data, the so-called Mooney plot transform. First, the small-to-moderate regime, where the polymeric chains unfold easily and the Mooney plot is almost linear. Second, the strain-hardening regime, where blobs of bundled chains unfold to stiffen the response in correspondence to the ‘upturn’ of the Mooney plot. Third, the limiting-chain regime, with a sharp stiffening occurring as the chains extend towards their limit. We provide strain-energy functions with terms accounting for each stage that (i) give an accurate local and then global fitting of the data; (ii) are consistent with weak nonlinear elasticity theory and (iii) can be interpreted in the framework of statistical mechanics. We apply our method to Treloar's classical experimental data and also to some more recent data. Our method not only provides models that describe the experimental data with a very low quantitative relative error, but also shows that the theory of nonlinear elasticity is much more robust that seemed at first sight.

Bubble dynamics in a viscoelastic medium with nonlinear elasticity

2015 ◽  
Vol 766 ◽  
pp. 54-75 ◽  
Author(s):  
R. Gaudron ◽  
M. T. Warnez ◽  
E. Johnsen
Keyword(s):  
Soft Tissue ◽  
Nonlinear Elasticity ◽  
Viscoelastic Medium ◽  
Strain Energy ◽  
Bubble Dynamics ◽  
Surrounding Medium ◽  
Viscoelastic Model ◽  
Energy Functions ◽  
Newtonian Liquids ◽  
Strain Energy Functions

AbstractIn a variety of recently developed medical procedures, bubbles are formed directly in soft tissue and may cause damage. While cavitation in Newtonian liquids has received significant attention, bubble dynamics in tissue, a viscoelastic medium, remains poorly understood. To model tissue, most previous studies have focused on Maxwell-based viscoelastic fluids. However, soft tissue generally possesses an original configuration to which it relaxes after deformation. Thus, a Kelvin–Voigt-based viscoelastic model is expected to be a more appropriate representation. Furthermore, large oscillations may occur, thus violating the infinitesimal strain assumption and requiring a nonlinear/finite-strain elasticity description. In this article, we develop a theoretical framework to simulate spherical bubble dynamics in a viscoelastic medium with nonlinear elasticity. Following modern continuum mechanics formalism, we derive the form of the elastic forces acting on a bubble for common strain-energy functions (e.g. neo-Hookean, Mooney–Rivlin) and incorporate them into Rayleigh–Plesset-like equations. The main effects of nonlinear elasticity are to reduce the violence of the collapse and rebound for large departures from the equilibrium radius, and increase the oscillation frequency. The present approach can readily be extended to other strain-energy functions and used to compute the stress/deformation fields in the surrounding medium.

On the Small Strain Behavior of Peroxide Crosslinked Natural Rubber

1986 ◽  
Vol 59 (1) ◽  
pp. 130-137 ◽  
Author(s):  
Gregory B. McKenna ◽  
Louis J. Zapas
Keyword(s):  
Energy Density ◽  
Density Function ◽  
Strain Energy Density ◽  
Strain Energy ◽  
Scaling Law ◽  
Small Strain ◽  
Constant Length ◽  
Strain Behavior ◽  
Small Strains ◽  
Strain Energy Density Function

Abstract Torque and normal force measurements on a cylinder subjected to torsion at constant length were used to study the behavior of NR crosslinked with 5 phr dicumyl peroxide. The derivatives of the strain-energy density function ∂W/∂I1 and ∂W/∂I2 were calculated from the data using the scaling law of Penn and Kearsley. The new results extend the limit of small strains at which the strain-energy density function derivatives have been measured to γ<0.005 and further confirm our previous results that for peroxide-crosslinked NR, ∂W/∂I2 does not become negative at small strain, contrary to several reports in the literature. Reduced stress was determined for the rubber by using the approach of Kearsley and Zapas to calculate the derivative w′(λ) of the Valanis-Landel form of the strain energy function. The results were compared with the measured values for reduced stress in tension and compression at small strains. While the deviation between the predictions and the experimental behavior do not exceed 6%, the characters of the calculated and measured reduced stress plots are different. The measurements in torsion were not obtained at small enough strains to enable direct comparison with the extension/compression behavior at |ε|<0.002. Extrapolation of the results did not produce the anomalous cusp observed in the reduced stress for 0.998<1/λ<1.002 which was reported in our previous study. The fact that torsional data do not show the cusp offers support to the Kearsley suggestion that at these extremely small deformations, rubber compressibility may play an important role in the stress-strain behavior. This could also explain the apparent discrepancy between the predicted Valanis-Landel behavior and the observed behavior. Future work involving higher precision experiments is required to resolve the matter.

Studies of quality of geometrically nonlinear elasticity theory for small strains and arbitrary displacements

2009 ◽  
Vol 44 (6) ◽  
pp. 837-851 ◽  
Author(s):  
D. V. Berezhnoi ◽  
V. N. Paimushin ◽  
V. I. Shalashilin
Keyword(s):  
Elasticity Theory ◽  
Nonlinear Elasticity ◽  
Geometrically Nonlinear ◽  
Nonlinear Elasticity Theory ◽  
Small Strains

A Note on the Pure Bending of Nonhomogeneous Prismatic Bars

2009 ◽  
Vol 37 (2) ◽  
pp. 118-129 ◽  
Author(s):  
Attila Baksa ◽  
István Ecsedi
Keyword(s):  
Composite Beam ◽  
Strain Energy ◽  
Elastic Strain Energy ◽  
Functionally Graded ◽  
Bending Stress ◽  
Pure Bending ◽  
Small Strains ◽  
Compressive Stresses ◽  
Elastic Bar

The paper examines the pure bending of a linearly elastic, isotropic, nonhomogeneous bar. The bending stress, elastic strain energy and the end cross-section rotations are determined for in-plane variation of the Young's modulus with small strains and displacements. It is shown that the governing formulae for elastic pure bending of nonhomogeneous bars have same forms as formulae for symmetrical bending (bending in the principal planes) of homogeneous bars. Two examples illustrate the application of the developed formulae. In the first, a composite beam is considered; the second deals with the determination of the maximum tensile and compressive stresses in a bent functionally graded elastic bar.

scholarly journals The stress field in a pulled cork and some subtle points in the semi-inverse method of nonlinear elasticity

2007 ◽  
Vol 463 (2087) ◽  
pp. 2945-2959 ◽  
Author(s):  
Riccardo De Pascalis ◽  
Michel Destrade ◽  
Giuseppe Saccomandi
Keyword(s):  
Nonlinear Elasticity ◽  
Strain Energy Density ◽  
Strain Energy ◽  
Inverse Method ◽  
Secondary Deformation ◽  
Normal Stress Differences ◽  
Shrink Fit ◽  
Incompressible Rubber ◽  
Asymptotic Analyses ◽  
Insight Into

In an attempt to describe cork-pulling, we model a cork as an incompressible rubber-like material and consider that it is subject to a helical shear deformation superimposed onto a shrink fit and a simple torsion. It turns out that this deformation field provides an insight into the possible appearance of secondary deformation fields for special classes of materials. We also find that these latent deformation fields are woken up by normal stress differences. We present some explicit examples based on the neo-Hookean, the generalized neo-Hookean and the Mooney–Rivlin forms of the strain-energy density. Using the simple exact solution found in the neo-Hookean case, we conjecture that it is advantageous to accompany the usual vertical axial force by a twisting moment, in order to extrude a cork from the neck of a bottle efficiently. Then we analyse departures from the neo-Hookean behaviour by exact and asymptotic analyses. In that process, we are able to give an elegant and analytic example of secondary (or latent) deformations in the framework of nonlinear elasticity.

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