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.

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.

scholarly journals Heuristic search for a predictive strain-energy function in nonlinear elasticity

2009 ◽  
Vol 46 (2) ◽  
pp. 271-286 ◽  
Author(s):  
Dmitri Miroshnychenko ◽  
W.A. Green
Keyword(s):  
Nonlinear Elasticity ◽  
Energy Function ◽  
Heuristic Search ◽  
Strain Energy ◽  
Strain Energy Function

scholarly journals Wrinkles and creases in the bending, unbending and eversion of soft sectors

2018 ◽  
Vol 474 (2212) ◽  
pp. 20170827 ◽  
Author(s):  
Taisiya Sigaeva ◽  
Robert Mangan ◽  
Luigi Vergori ◽  
Michel Destrade ◽  
Les Sudak
Keyword(s):  
Nonlinear Elasticity ◽  
Strain Energy ◽  
Strong Ellipticity ◽  
Science And Engineering ◽  
Third Order ◽  
Ellipticity Condition ◽  
Weakly Nonlinear ◽  
Curved Structures ◽  
Incompressible Hyperelastic Material ◽  
Strong Ellipticity Condition

We study what is clearly one of the most common modes of deformation found in nature, science and engineering, namely the large elastic bending of curved structures, as well as its inverse, unbending, which can be brought beyond complete straightening to turn into eversion. We find that the suggested mathematical solution to these problems always exists and is unique when the solid is modelled as a homogeneous, isotropic, incompressible hyperelastic material with a strain-energy satisfying the strong ellipticity condition. We also provide explicit asymptotic solutions for thin sectors. When the deformations are severe enough, the compressed side of the elastic material may buckle and wrinkles could then develop. We analyse, in detail, the onset of this instability for the Mooney–Rivlin strain energy, which covers the cases of the neo-Hookean model in exact nonlinear elasticity and of third-order elastic materials in weakly nonlinear elasticity. In particular, the associated theoretical and numerical treatment allows us to predict the number and wavelength of the wrinkles. Guided by experimental observations, we finally look at the development of creases, which we simulate through advanced finite-element computations. In some cases, the linearized analysis allows us to predict correctly the number and the wavelength of the creases, which turn out to occur only a few per cent of strain earlier than the wrinkles.

The wustite-spinel interface

1987 ◽  
Vol 45 ◽  
pp. 268-269
Author(s):  
S.R. Summerfelt ◽  
C.B. Carter
Keyword(s):  
Solid State ◽  
Solid State Reaction ◽  
Strain Energy ◽  
Matrix Material ◽  
Lattice Misfit ◽  
Solid State Reactions ◽  
Oxygen Sublattice ◽  
Large Particles ◽  
Small Lattice ◽  
Coherent Interfaces

The wustite-spinel interface can be viewed as a model interface because the wustite and spinel can share a common f.c.c. oxygen sublattice such that only the cations distribution changes on crossing the interface. In this study, the interface has been formed by a solid state reaction involving either external or internal oxidation. In systems with very small lattice misfit, very large particles (>lμm) with coherent interfaces have been observed. Previously, the wustite-spinel interface had been observed to facet on {111} planes for MgFe2C4 and along {100} planes for MgAl2C4 and MgCr2O4, the spinel then grows preferentially in the <001> direction. Reasons for these experimental observations have been discussed by Henriksen and Kingery by considering the strain energy. The point-defect chemistry of such solid state reactions has been examined by Schmalzried. Although MgO has been the principal matrix material examined, others such as NiO have also been studied.

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