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.

Elasticity M-tensors and the strong ellipticity condition

2020 ◽  
Vol 373 ◽  
pp. 124982 ◽  
Author(s):  
Weiyang Ding ◽  
Jinjie Liu ◽  
Liqun Qi ◽  
Hong Yan
Keyword(s):  
Strong Ellipticity ◽  
Ellipticity Condition ◽  
Strong Ellipticity Condition

THE RATE OF CONVERGENCE AND WEAKER CONVERGENT CONDITION FOR THE METHOD FOR FINDING THE LARGEST SINGULAR VALUE OF RECTANGULAR TENSORS

2016 ◽  
Vol 78 (6-5) ◽  
Author(s):  
Nur Fadhilah Ibrahim ◽  
Nurul Akmal Mohamed
Keyword(s):  
Quantum Physics ◽  
Solid Mechanics ◽  
Linear Convergence ◽  
Singular Value ◽  
Strong Ellipticity ◽  
Ellipticity Condition ◽  
Condition Problem ◽  
Strong Ellipticity Condition ◽  
Weak Irreducibility ◽  
Rectangular Tensors

The applications of real rectangular tensors, among others, including the strong ellipticity condition problem within solid mechanics, and the entanglement problem within quantum physics. A method was suggested by Zhou, Caccetta and Qi in 2013, as a means of calculating the largest singular value of a nonnegative rectangular tensor. In this paper, we show that the method converges under weak irreducibility condition, and that it has a Q-linear convergence.   

scholarly journals On Ellipticity of Hyperelastic Models Based on Experimental Data

2017 ◽  
Vol 63 (3) ◽  
pp. 504-515
Author(s):  
V Yu Salamatova ◽  
Yu V Vasilevskii
Keyword(s):  
Experimental Data ◽  
Mechanical Behavior ◽  
Equilibrium Equation ◽  
Deformation Gradient ◽  
Necessary Condition ◽  
Strong Ellipticity ◽  
Correct Description ◽  
Ellipticity Condition ◽  
Strong Ellipticity Condition ◽  
Hyperelastic Models

The condition of ellipticity of the equilibrium equation plays an important role for correct description of mechanical behavior of materials and is a necessary condition for new defining relationships. Earlier, new deformation measures were proposed to vanish correlations between the terms, that dramatically simplifies restoration of defining relationships from experimental data. One of these new deformation measures is based on the QR-expansion of deformation gradient. In this paper, we study the strong ellipticity condition for hyperelastic material using the QR-expansion of deformation gradient.

scholarly journals Buckling of a spinning elastic cylinder: linear, weakly nonlinear and post-buckling analyses

2018 ◽  
Vol 474 (2216) ◽  
pp. 20180242 ◽  
Author(s):  
Franck Richard ◽  
Aditi Chakrabarti ◽  
Basile Audoly ◽  
Yves Pomeau ◽  
Serge Mora
Keyword(s):  
Strain Energy ◽  
Elastic Cylinder ◽  
Hyperelastic Material ◽  
Quadratic Term ◽  
Buckling Modes ◽  
Weakly Nonlinear ◽  
Incompressible Hyperelastic Material ◽  
Post Buckling ◽  
Equivalent Shear Modulus ◽  
Nonlinear Regimes

An elastic cylinder spinning about a rigid axis buckles beyond a critical angular velocity, by an instability driven by the centrifugal force. This instability and the competition between the different buckling modes are investigated using analytical calculations in the linear and weakly nonlinear regimes, complemented by numerical simulations in the fully post-buckled regime. The weakly nonlinear analysis is carried out for a generic incompressible hyperelastic material. The key role played by the quadratic term in the expansion of the strain energy density is pointed out: this term has a strong effect on both the nature of the bifurcation, which can switch from supercritical to subcritical, and the buckling amplitude. Given an arbitrary hyperelastic material, an equivalent shear modulus is proposed, allowing the main features of the instability to be captured by an equivalent neo-Hookean model.

scholarly journals Sharp bounds on the minimum $M$-eigenvalue and strong ellipticity condition of elasticity $Z$-tensors-tensors

2021 ◽  
Vol 0 (0) ◽  
pp. 0
Author(s):  
Chong Wang ◽  
Gang Wang ◽  
Lixia Liu
Keyword(s):  
Lower Bound ◽  
Lower Bounds ◽  
Necessary Conditions ◽  
Upper And Lower Bounds ◽  
Symmetric Matrices ◽  
Strong Ellipticity ◽  
Ellipticity Condition ◽  
Numerical Examples ◽  
Extreme Eigenvalue ◽  
Strong Ellipticity Condition

<p style='text-indent:20px;'>In this paper, we establish sharp upper and lower bounds on the minimum <i>M</i>-eigenvalue via the extreme eigenvalue of the symmetric matrices extracted from elasticity <i>Z</i>-tensors without irreducible conditions. Based on the lower bound estimations for the minimum <i>M</i>-eigenvalue, we provide some checkable sufficient or necessary conditions for the strong ellipticity condition. Numerical examples are given to demonstrate the proposed results.</p>

Variational formulation for weakly nonlinear perturbations of ideal magnetohydrodynamics

2011 ◽  
Vol 77 (5) ◽  
pp. 589-615 ◽  
Author(s):  
M. HIROTA
Keyword(s):  
Coupling Coefficient ◽  
Displacement Vector ◽  
Lagrange Equation ◽  
Natural Extension ◽  
Equation Of Motion ◽  
Nonlinear Phenomena ◽  
Lie Series ◽  
Third Order ◽  
Weakly Nonlinear ◽  
Ideal Magnetohydrodynamics

AbstractA new equation of motion that governs weakly nonlinear phenomena in ideal magnetohydrodynamics (MHDs) is derived as a natural extension of the well-known linearized equation of motion for the displacement field. This derivation is made possible by expanding the MHD Lagrangian explicitly up to third order with respect to the displacement of plasma, which necessitates an efficient use of the Lie series expansion. The resultant equation of motion (i.e. the Euler–Lagrange equation) includes a new quadratic force term which is responsible for various mode–mode coupling due to the MHD nonlinearity. The third-order potential energy serves to quantify the coupling coefficient among resonant three modes and its cubic symmetry proves the Manley–Rowe relations. In contrast to earlier works, the coupling coefficient is expressed only by the displacement vector field, which is already familiar in the linear MHD theory, and both the fixed and free boundary cases are treated systematically.

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