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.

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.   

On the Ability of Structural and Phenomenological Hyperelastic Models to Predict the Mechanical Behavior of Biological Tissues Submitted to Multiaxial Loadings

2012 ◽  
Author(s):  
Mathieu Nierenberger ◽  
Yves Rémond ◽  
Saïd Ahzi
Keyword(s):  
Experimental Data ◽  
Mechanical Behavior ◽  
Strain Energy ◽  
Soft Tissues ◽  
Least Squares Method ◽  
Biological Tissues ◽  
Physical Parameters ◽  
Cauchy Stress ◽  
Specific Loading ◽  
Hyperelastic Models

Medical surgery is currently rapidly improving and requires modeling faithfully the mechanical behavior of soft tissues. Various models exist in literature; some of them created for the study of biological materials, and others coming from the field of rubber mechanics. Indeed biological tissues show a mechanical behavior close to the one of rubbers. But while building a model, one has to keep in mind that its parameters should be loading independent and that the model should be able to predict the behavior under complex loading conditions. In addition, keeping physical parameters seems interesting since it allows a bottom up approach taking into account the microstructure of the material. In this study, the authors consider different existing hyperelastic models based on strain energy functions and identify their coefficients successively on single loading stress-stretch curves. The experimental data used, come from a paper by Zemanek dated 2009 and concerning uniaxial, equibiaxial and plane tension tests on porcine arterial walls taken in identical experimental conditions. To achieve identification, the strain energy function of each model is derived differently to provide an expression of the Cauchy stress associated to each loading case. Firstly the parameters of each model are identified on the uniaxial tension curve using a least squares method. Then, keeping the obtained parameters, predictions are made for the two other loading cases (equibiaxial and plane tension) using the associated expressions of stresses. A comparison of these predictions with experimental data is done and allows evaluating the predictive capabilities of each model for the different loading cases. A similar approach is used after swapping the loading types. Since the predictive capabilities of the models are really dependent on the loading chosen to determine their parameters, another type of identification procedure is set up. It consists in adding the residues over the three loading cases during identification. This alternative identification method allows a better agreement between each model and the various types of experiments. This study evaluated the ability of some classical hyperelastic models to be used for a predictive scope after being identified on a specific loading type. Besides it brought to light some existing models which can describe at best the mechanical behavior of biological tissues submitted to various loadings.

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>

scholarly journals Polyconvex hyperelastic modeling of rubberlike materials

2021 ◽  
Vol 43 (7) ◽  
Author(s):  
Cyprian Suchocki ◽  
Stanisław Jemioło
Keyword(s):  
Experimental Data ◽  
Power Law ◽  
Curve Fitting ◽  
Constitutive Relations ◽  
Data Sets ◽  
Power Law Model ◽  
Stored Energy ◽  
Data Approximation ◽  
Exponential Power ◽  
Hyperelastic Models

AbstractIn this work a number of selected, isotropic, invariant-based hyperelastic models are analyzed. The considered constitutive relations of hyperelasticity include the model by Gent (G) and its extension, the so-called generalized Gent model (GG), the exponential-power law model (Exp-PL) and the power law model (PL). The material parameters of the models under study have been identified for eight different experimental data sets. As it has been demonstrated, the much celebrated Gent’s model does not always allow to obtain an acceptable quality of the experimental data approximation. Furthermore, it is observed that the best curve fitting quality is usually achieved when the experimentally derived conditions that were proposed by Rivlin and Saunders are fulfilled. However, it is shown that the conditions by Rivlin and Saunders are in a contradiction with the mathematical requirements of stored energy polyconvexity. A polyconvex stored energy function is assumed in order to ensure the existence of solutions to a properly defined boundary value problem and to avoid non-physical material response. It is found that in the case of the analyzed hyperelastic models the application of polyconvexity conditions leads to only a slight decrease in the curve fitting quality. When the energy polyconvexity is assumed, the best experimental data approximation is usually obtained for the PL model. Among the non-polyconvex hyperelastic models, the best curve fitting results are most frequently achieved for the GG model. However, it is shown that both the G and the GG models are problematic due to the presence of the locking effect.

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