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.   

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

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 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.

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>

Singular value inclusion sets of rectangular tensors

2019 ◽  
Vol 576 ◽  
pp. 181-199 ◽  
Author(s):  
Hongmei Yao ◽  
Can Zhang ◽  
Lei Liu ◽  
Jiang Zhou ◽  
Changjiang Bu
Keyword(s):  
Singular Value ◽  
Rectangular Tensors

scholarly journals An S-type upper bound for the largest singular value of nonnegative rectangular tensors

2016 ◽  
Vol 14 (1) ◽  
pp. 925-933 ◽  
Author(s):  
Jianxing Zhao ◽  
Caili Sang
Keyword(s):  
Upper Bound ◽  
Singular Value ◽  
Numerical Examples ◽  
Rectangular Tensors ◽  
Theoretical Results

AbstractAn S-type upper bound for the largest singular value of a nonnegative rectangular tensor is given by breaking N = {1, 2, … n} into disjoint subsets S and its complement. It is shown that the new upper bound is smaller than that provided by Yang and Yang (2011). Numerical examples are given to verify the theoretical results.

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