scholarly journals Likely oscillatory motions of stochastic hyperelastic solids

2019 ◽  
Vol 3 (1) ◽  
Author(s):  
L Angela Mihai ◽  
Danielle Fitt ◽  
Thomas E Woolley ◽  
Alain Goriely
Keyword(s):  
Strain Energy ◽  
Probability Distributions ◽  
Time Instant ◽  
Spherical Shells ◽  
Mechanical Responses ◽  
Elastic Systems ◽  
Cylindrical Tubes ◽  
Parameter Interval ◽  
Isotropic Hyperelastic Material ◽  
Quantities Of Interest

Abstract Stochastic homogeneous hyperelastic solids are characterized by strain-energy densities where the parameters are random variables defined by probability density functions. These models allow for the propagation of uncertainties from input data to output quantities of interest. To investigate the effect of probabilistic parameters on predicted mechanical responses, we study radial oscillations of cylindrical and spherical shells of stochastic incompressible isotropic hyperelastic material, formulated as quasi-equilibrated motions where the system is in equilibrium at every time instant. Additionally, we study finite shear oscillations of a cuboid, which are not quasi-equilibrated. We find that, for hyperelastic bodies of stochastic neo-Hookean or Mooney–Rivlin material, the amplitude and period of the oscillations follow probability distributions that can be characterized. Further, for cylindrical tubes and spherical shells, when an impulse surface traction is applied, there is a parameter interval where the oscillatory and non-oscillatory motions compete, in the sense that both have a chance to occur with a given probability. We refer to the dynamic evolution of these elastic systems, which exhibit inherent uncertainties due to the material properties, as ‘likely oscillatory motions’.

On the Hyperelastic Pressurized Thick-Walled Spherical Shells and Cylindrical Tubes Using the Analytical Closed-Form Solutions

2015 ◽  
Vol 07 (02) ◽  
pp. 1550027 ◽  
Author(s):  
D. M. Taghizadeh ◽  
A. Bagheri ◽  
H. Darijani
Keyword(s):  
Closed Form ◽  
Strain Energy ◽  
Soft Tissues ◽  
Behavior Modeling ◽  
Type Model ◽  
Mathematical Form ◽  
Spherical Shells ◽  
Closed Form Solutions ◽  
Axial Forces ◽  
Cylindrical Tubes

This paper focuses on thick-walled spherical shells and cylindrical tubes made of the soft tissues and the rubber-like materials. These materials are characterized by high deformability in which their stress–stretch curves are arranged in the range of S-shaped to J-shaped forms. From the continuum viewpoint, a strain energy density function is postulated for modeling the behavior of these materials. In order to fulfill the main aims of this paper, among all existing energy functions including polynomial, power law, logarithmic and exponential functions, or a linear combination of them, we deduced to evaluate the performance of an Ogden-type model with only integer powers for the mechanical behavior modeling of the S-shaped to J-shaped materials. Most of all, this strain energy function because of its mathematical form can play a constructive role in presentation of the analytical closed-form solutions for the boundary value problems in the field of the finite deformation elasticity. This constitutive model due to the high performance in constitutive modeling and the simplicity of its mathematical form is applied to pressurized thick-walled spherical shells and cylindrical tubes in order to find a closed-form analytical solution for their analysis. Using these analytical solutions, a comprehensive study is done on vanishing circumstance of the snap-through instability that occurs in the inflation of internally pressurized spherical shells and cylindrical tubes. It was observed that the parameters such as shell thickness, the elastic material properties specially the materials with J-shaped mechanical behaviors and the absence and presence of axial forces in cylindrical tubes have significant influence on vanishing of the snap-through instability in the thick-walled pressurized spherical shells and cylindrical tubes.

scholarly journals Modelling the Inflation and Elastic Instabilities of Rubber-Like Spherical and Cylindrical Shells Using a New Generalised Neo-Hookean Strain Energy Function

2021 ◽  
Author(s):  
Afshin Anssari-Benam ◽  
Andrea Bucchi ◽  
Giuseppe Saccomandi
Keyword(s):  
Experimental Data ◽  
Energy Function ◽  
Limit Point ◽  
Strain Energy ◽  
Cylindrical Shells ◽  
Strain Energy Function ◽  
Model Parameters ◽  
Wide Range ◽  
Cylindrical Tubes ◽  
Gent Model

AbstractThe application of a newly proposed generalised neo-Hookean strain energy function to the inflation of incompressible rubber-like spherical and cylindrical shells is demonstrated in this paper. The pressure ($P$ P ) – inflation ($\lambda $ λ or $v$ v ) relationships are derived and presented for four shells: thin- and thick-walled spherical balloons, and thin- and thick-walled cylindrical tubes. Characteristics of the inflation curves predicted by the model for the four considered shells are analysed and the critical values of the model parameters for exhibiting the limit-point instability are established. The application of the model to extant experimental datasets procured from studies across 19th to 21st century will be demonstrated, showing favourable agreement between the model and the experimental data. The capability of the model to capture the two characteristic instability phenomena in the inflation of rubber-like materials, namely the limit-point and inflation-jump instabilities, will be made evident from both the theoretical analysis and curve-fitting approaches presented in this study. A comparison with the predictions of the Gent model for the considered data is also demonstrated and is shown that our presented model provides improved fits. Given the simplicity of the model, its ability to fit a wide range of experimental data and capture both limit-point and inflation-jump instabilities, we propose the application of our model to the inflation of rubber-like materials.

Hydrolytic Aging in Rubber-Like Materials: A Micro-Mechanical Approach to Modeling

2019 ◽  
Author(s):  
Amir Bahrololoumi ◽  
Roozbeh Dargazany
Keyword(s):  
Strain Energy ◽  
Chain Scission ◽  
Good Choice ◽  
Mullins Effect ◽  
Cross Link ◽  
Original Matrix ◽  
Mechanical Responses ◽  
Permanent Set ◽  
Mechanical Approach ◽  
Fitting Parameters

Abstract The effect of hydrolytic aging on mechanical responses of Rubber likes materials, in particular, Mullins effect and the permanent set has been modeled. Hydrolytic aging is considered as the result of the competition between two phenomena (1) chain scission and (2) cross-link scission/reformation. Both phenomena were modeled and thus, the strain energy of the polymer matrix is written with respect to three independent mechanisms; i) the shrinking original matrix which has not been attacked by water, ii) conversion of the first network to a new network due to the reduction of the crosslinks, and iii) energy loss from network degradation due to water attacks to ester groups. The model is validated with respect to a set of experimental data. Besides accuracy, the simplicity and few numbers of fitting parameters make the model a good choice for further implementations.

Photonic band gap in the face-centered cubic lattice of spherical shells connected by cylindrical tubes

2005 ◽  
Vol 72 (11) ◽  
Author(s):  
Hong-Bo Chen ◽  
Yong-Zheng Zhu ◽  
Yan-Ling Cao ◽  
Yan-Ping Wang ◽  
Yuan-Bin Chi
Keyword(s):  
Band Gap ◽  
Photonic Band Gap ◽  
Face Centered Cubic ◽  
Spherical Shells ◽  
Cubic Lattice ◽  
Cylindrical Tubes ◽  
The Face ◽  
Face Centered ◽  
Photonic Band

scholarly journals Probabilistic Concepts in a Changing Climate: A Snapshot Attractor Picture*

2015 ◽  
Vol 28 (8) ◽  
pp. 3275-3288 ◽  
Author(s):  
Gábor Drótos ◽  
Tamás Bódai ◽  
Tamás Tél
Keyword(s):  
Large Scale ◽  
Climate Model ◽  
Probability Distributions ◽  
Internal Variability ◽  
Time Instant ◽  
Convergence Time ◽  
Natural Distribution ◽  
Changing Climate ◽  
Temperature Contrast

Abstract The authors argue that the concept of snapshot attractors and of their natural probability distributions are the only available tools by means of which mathematically sound statements can be made about averages, variances, etc., for a given time instant in a changing climate. A basic advantage of the snapshot approach, which relies on the use of an ensemble, is that the natural distribution and thus any statistics based on it are independent of the particular ensemble used, provided it is initiated in the past earlier than a convergence time. To illustrate these concepts, a tutorial presentation is given within the framework of a low-order model in which the temperature contrast parameter over a hemisphere decreases linearly in time. Furthermore, the averages and variances obtained from the snapshot attractor approach are demonstrated to strongly differ from the traditional 30-yr temporal averages and variances taken along single realizations. The authors also claim that internal variability can be quantified by the natural distribution since it characterizes the chaotic motion represented by the snapshot attractor. This experience suggests that snapshot-attractor-based calculations might be appropriate to be evaluated in any large-scale climate model, and that the application of 30-yr temporal averages taken along single realizations should be complemented with this more appealing tool for the characterization of climate changes, which seems to be practically feasible with moderate ensemble sizes.

scholarly journals Anomalous strain energy transformation pathways in mechanical metamaterials

2019 ◽  
Vol 475 (2226) ◽  
pp. 20190041 ◽  
Author(s):  
Eduard G. Karpov ◽  
Larry A. Danso ◽  
John T. Klein
Keyword(s):  
Strain Energy ◽  
Spectral Energy Distribution ◽  
Spectral Energy ◽  
Surface Load ◽  
Simple Relationship ◽  
Spectral Entropy ◽  
Mechanical Responses ◽  
Simple Lattice ◽  
Lattice Materials ◽  
Mechanical Metamaterials

This discussion starts with a mechanics version of Parseval's energy theorem applicable to any discrete lattice material with periodic internal structure: a microtruss, grid, frame, origami or tessellation. It provides a simple relationship between the strain energy volumetric/usual and spectral distributions in the reciprocal space. The spectral energy distribution leads directly to a spectral entropy of lattice deformation (Shannon's type), whose variance with a material coordinate represents the decrease of information about surface loads in the material interior. Spectral entropy is also a basic measure of complexity of mechanical responses of metamaterials to surface and body loads. Considering transformation of the energy volumetric and spectral distributions with a material coordinate pointed away from a surface load, several interesting anomalies are seen even for simple lattice materials, when compared to continuum materials. These anomalies include selective filtering of surface Raleigh waves (sinusoidal pressure patterns), Saint–Venant effect inversion illustrated by energy spectral distribution contours, occurrence of ‘hiding pockets’ of low deformation, and redirection of strain energy maximum away from axis of a concentrated surface load. The latter phenomenon can be significant for impact protection applications of mechanical metamaterials.

MECHANICAL BEHAVIOUR OF TRANSVERSELY ISOTROPIC POROUS NEO-HOOKEAN SOLIDS

2010 ◽  
Vol 02 (01) ◽  
pp. 11-39 ◽  
Author(s):  
ZAOYANG GUO ◽  
FERHUN C. CANER
Keyword(s):  
Strain Energy ◽  
Hydrostatic Stress ◽  
Strain Energy Function ◽  
Transversely Isotropic ◽  
Axial Direction ◽  
Shear Loading ◽  
Porous Solid ◽  
Mechanical Responses ◽  
Cylindrical Pores ◽  
Hyperelastic Model

In this paper, the mechanical responses of a recently developed hyperelastic model for the neo-Hookean solids with aligned continuous cylindrical pores under finite homogeneous deformation that can capture the anisotropic compressibility as well as the coupling between the volumetric and deviatoric behaviours are examined. To this end, the strain energy function of this hyperelastic compressible transversely isotropic model contains terms for the coupling of volumetric and deviatoric behaviours. It is shown that, the asymptotic response of this anisotropic compressible model under extreme loading situations is considerably different from that of incompressible models. The unstable behaviour of the porous solid under hydrostatic stress/strain loadings is discussed in detail. When a general simple 2D shear deformation is applied to this porous solid in i1 – i2 plane, the normal stress in the third axial direction (i3) is nonzero. The loss of monotonicity of the stress tensor under off-axis simple 2D shear loading is demonstrated as well.

The need to account for residual strains and composite nature of heart wall in mechanical analyses

1996 ◽  
Vol 271 (3) ◽  
pp. H947-H961 ◽  
Author(s):  
T. Kang ◽  
F. C. Yin
Keyword(s):  
Strain Energy ◽  
Left Ventricular ◽  
Energy Functions ◽  
Mechanical Responses ◽  
Residual Strains ◽  
Composite Nature ◽  
Strain Energy Functions ◽  
Heart Wall ◽  
Strain Responses ◽  
Canine Myocardium

We studied 19 excised, passive rabbit left ventricular walls to delineate the forms of the strain-energy functions (W) for myocardium and epicardium, to quantify residual strains across the wall, and to investigate whether the mechanical behavior of the intact wall can be predicted by accounting for the above properties. The unloaded dimensions and the stress-strain responses to equibiaxial and uniaxial loadings were obtained first for the intact wall and then individually for the epicardium and myocardium. Results show that the previously proposed W for canine myocardium and epicardium are suitable. The unloaded intact wall has residual strains: the epicardium is stretched and the myocardium shrunk from their respective isolated, unloaded states. The predicted mechanical responses of the intact wall to biaxial loadings were inaccurate when the residual strains were not taken into account. Accounting for these, however, yielded reasonable predictions. Thus information on the unloaded reference state and properties of each portion is needed to accurately predict the behavior of the intact wall.

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