A Preliminary Analysis of the Data From an in Vitro Inflation-Extension Test Can Validate the Assumption of Arterial Tissue Elasticity

2013 ◽  
Vol 135 (8) ◽  
Author(s):  
Alexander Rachev ◽  
Tarek Shazly
Keyword(s):  
Experimental Data ◽  
Renal Artery ◽  
Strain Energy ◽  
Mechanical Response ◽  
Strain Energy Function ◽  
Elastic Solid ◽  
Tissue Elasticity ◽  
Biaxial Stretching ◽  
Constitutive Formulation

The objective of this study is to propose a method for preliminary processing of the experimental data from an inflation-extension test on tubular arterial specimens. The method is based on the condition for existence of a strain energy function (SEF) and can be used to verify whether the data from a certain experiment validate the assumption that the tissue can be considered as an elastic solid. As an illustrative example of the proposed method, experimental data for a porcine renal artery are used and the sources of the error in satisfying the condition of elasticity are analyzed. The results lead to the conclusion that the experimental data for a renal artery validate that the artery exhibits an elastic mechanical response and a constitutive formulation based on the existence of the SEF is justified. A modification of the proposed method for the case of an in-plane biaxial stretching test of mechanically isotropic and orthotropic tissues is considered.

A Strain Energy Function for the Annulus Fibrosus Specified Using Biaxial Experimental Data

1999 ◽  
Author(s):  
Elisa C. Bass ◽  
Jeffrey C. Lotz
Keyword(s):  
Energy Function ◽  
Annulus Fibrosus ◽  
Strain Energy ◽  
Mechanical Response ◽  
Biaxial Tension ◽  
Elastic Strain Energy ◽  
Strain Energy Function ◽  
Constitutive Formulation

Abstract The mechanical behavior of the annulus fibrosus has typically been characterized through the use of uniaxial tests. In contrast, its in vivo constraints are multiaxial and likely result in a mechanical response very different from that observed to date in vitro. The goal of this study was to test the annulus in biaxial tension and use these data to determine an elastic strain energy function for the annulus. Our results demonstrate that the mechanical response of the annulus is dramatically influenced by a biaxial constraint, and that these experiments provide important data for the determination of the constitutive formulation for this strongly anisotropic and nonlinear tissue.

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.

Three-Dimensional Stress Distribution in Arteries

1983 ◽  
Vol 105 (3) ◽  
pp. 268-274 ◽  
Author(s):  
C. J. Chuong ◽  
Y. C. Fung
Keyword(s):  
Experimental Data ◽  
Stress Distribution ◽  
Vessel Wall ◽  
Strain Energy ◽  
Arterial Wall ◽  
Three Dimensional ◽  
Strain Energy Function ◽  
Exponential Form ◽  
Strain Relationship ◽  
Longitudinal Stretch

A three-dimensional stress-strain relationship derived from a strain energy function of the exponential form is proposed for the arterial wall. The material constants are identified from experimental data on rabbit arteries subjected to inflation and longitudinal stretch in the physiological range. The objectives are: 1) to show that such a procedure is feasible and practical, and 2) to call attention to the very large variations in stresses and strains across the vessel wall under the assumptions that the tissue is incompressible and stress-free when all external load is removed.

Analytical and Experimental Study of Beam Torsional Stiffness With Large Axial Elongation

1988 ◽  
Vol 55 (1) ◽  
pp. 171-178 ◽  
Author(s):  
M. Degener ◽  
D. H. Hodges ◽  
D. Petersen
Keyword(s):  
Experimental Data ◽  
Axial Force ◽  
Energy Function ◽  
Strain Energy ◽  
Strain Energy Function ◽  
Torsional Stiffness ◽  
Deformation Tensor ◽  
Axial Elongation ◽  
Green Strain ◽  
Strain Components

The axial force and effective torsional stiffness versus axial elongation are investigated analytically and experimentally for a beam of circular cross section and made of an incompressible material that can sustain large elastic deformation. An approach based on a strain energy function identical to that used in linear elasticity, except with its strain components replaced by those of some finite-deformation tensor, would be expected to provide only limited predictive capability for this large-strain problem. Indeed, such an approach based on Green strain components (commonly referred to as the geometrically nonlinear theory of elasticity) incorrectly predicts a change in volume and predicts the wrong trend regarding the experimentally determined axial force and effective torsional stiffness. On the other hand, use of the same strain energy function, only with the Hencky logarithmic strain components, correctly predicts constant volume and provides excellent agreement with experimental data for lateral contraction, tensile force, and torsional stiffness—even when the axial elongation is large. For strain measures other than Hencky, the strain energy function must be modified to consistently account for large strains. For comparison, theoretical curves derived from a modified Green strain energy function are added. This approach provides results identical to those of the Neo-Hookean formulation for incompressible materials yielding fair agreement with the experimental results for coupled tension and torsion. An alternative approach, proposed in the present paper and based on a modified Almansi strain energy function, provides very good agreement with experimental data and is somewhat easier to manage than the Hencky strain energy approach.

Small Indentations of Rubber Blocks: Effect of Size and Shape of Block and of Lateral Compression

2006 ◽  
Vol 79 (4) ◽  
pp. 674-693 ◽  
Author(s):  
A. N. Gent ◽  
O. H. Yeoh
Keyword(s):  
Cross Section ◽  
Strain Energy ◽  
Rectangular Cross Section ◽  
Strain Energy Function ◽  
Elastic Solid ◽  
Biaxial Compression ◽  
Rigid Surface ◽  
Lateral Compression ◽  
Block Width ◽  
Rubber Block

Abstract Many gaskets and seals consist of a long rubber strip or thin-walled ring, placed on a flat rigid surface and indented by a flat-ended rigid indenter. We have examined their resistance to small indentations by FEA. They are treated as infinitely-long elastic blocks of rectangular cross-section, resting on a rigid frictionless base. The indentation stiffness is calculated for various ratios of indenter tip width to block width and to block thickness, using two restraint conditions on the outer surfaces: frictionless walls (zero outwards displacement), as for a gasket placed in a recess; or stress-free, as for a gasket with no lateral restraint. For an infinitely-wide and infinitely-thick block, the theoretical resistance to indentation is zero. For comparison, the indentation stiffness is calculated for cylindrical rubber blocks of varied radius and thickness, indented by a flat-ended cylindrical indenter. In this case the result for an infinitely-large block is finite. A second study treats indentation of a rubber block, pre-compressed in the surface plane. Biot showed that the indentation stiffness of a half-space becomes zero at a critical compression, about 33% for equi-biaxial compression and 44 % for plane strain compression, for both a neo-Hookean and a Mooney-Rivlin elastic solid. FEA calculations were made of the indentation stiffness of neo-Hookean blocks of various sizes, pre-compressed to various degrees. The results are compared with Biot's result. In an Appendix, the critical degree of compression is calculated for a more realistic strain energy function than either the neo-Hookean or the Mooney-Rivlin approximation.

Large Deformation Isotropic Elasticity—On the Correlation of Theory and Experiment for Incompressible Rubberlike Solids

1973 ◽  
Vol 46 (2) ◽  
pp. 398-416 ◽  
Author(s):  
R. W. Ogden
Keyword(s):  
Mathematical Analysis ◽  
Energy Function ◽  
Strain Energy ◽  
Mechanical Response ◽  
Strain Energy Function ◽  
Usual Practice ◽  
Principal Axes ◽  
Simple Tension ◽  
Isotropic Elasticity ◽  
Strain Invariants

Abstract Many attempts have been made to reproduce theoretically the stress-strain curves obtained from experiments on the isothermal deformation of highly elastic ‘rubberlike’ materials. The existence of a strain-energy function has usually been postulated, and the simplifications appropriate to the assumptions of isotropy and incompressibility have been exploited. However, the usual practice of writing the strain energy as a function of two independent strain invariants has, in general, the effect of complicating the associated mathematical analysis (this is particularly evident in relation to the calculation of instantaneous moduli of elasticity) and, consequently, the basic elegance and simplicity of isotropic elasticity is sacrificed. Furthermore, recently proposed special forms of the strain-energy function are rather complicated functions of two invariants. The purpose of this paper is, while making full use of the inherent simplicity of isotropic elasticity, to construct a strain-energy function which: (i) provides an adequate representation of the mechanical response of rubberlike solids, and (ii) is simple enough to be amenable to mathematical analysis. A strain-energy function which is a linear combination of strain invariants defined by ϕ(α)=(α1α+α2α+α3α)/α is proposed; and the principal stretches α1, α2, and α3 are used as independent variables subject to the incompressibility constraint α1α2α3=1. Principal axes techniques are used where appropriate. An excellent agreement between this theory and the experimental data from simple tension, pure shear and equibiaxial tension tests is demonstrated. It is also shown that the present theory has certain repercussions in respect of the constitutive inequality proposed by Hill.

Finding the Constitutive Relation for a Specific Elastomer

1993 ◽  
Vol 115 (3) ◽  
pp. 329-336 ◽  
Author(s):  
Yun Ling ◽  
Peter A. Engel ◽  
Wm. L. Brodskey ◽  
Yifan Guo
Keyword(s):  
Experimental Data ◽  
Energy Function ◽  
Strain Energy ◽  
Pure Shear ◽  
Strain Energy Function ◽  
Invariant Polynomial ◽  
Energy Functions ◽  
Finite Element Program ◽  
Biaxial Extension ◽  
Strain Energy Functions

The main purpose of this study was to determine a suitable strain energy function for a specific elastomer. A survey of various strain energy functions proposed in the past was made. For natural rubber, there were some specific strain energy functions which could accurately fit the experimental data for various types of deformations. The process of determining a strain energy function for the specific elastomer was then described. The second-order invariant polynomial strain energy function (James et al., 1975) was found to give a good fit to the experimental data of uniaxial tension, uniaxial compression, equi-biaxial extension, and pure shear. A new form of strain energy function was proposed; it yielded improved results. The equi-biaxial extension experiment was done in a novel way in which the moire techniques (Pendleton, 1989) were used. The obtained strain energy functions were then utilized in a finite element program to calculate the load-deflection relation of an electrometric spring used in an electrical connector.

scholarly journals Evolution of anisotropy in soft tissue

2014 ◽  
Vol 470 (2161) ◽  
pp. 20130548 ◽  
Author(s):  
J. G. Murphy
Keyword(s):  
Linear Theory ◽  
Energy Function ◽  
Anisotropic Material ◽  
Strain Energy ◽  
Mechanical Response ◽  
Essential Feature ◽  
Sufficient Conditions ◽  
Strain Energy Function ◽  
Phenomenological Approach ◽  
Reduced Form

The phenomenological approach to the modelling of the mechanical response of arteries usually assumes a reduced form of the strain-energy function in order to reduce the mathematical complexity of the model. A common approach eschews the full basis of seven invariants for the strain-energy function in favour of a reduced set of only three invariants. It is shown that this reduced form is not consistent with the corresponding full linear theory based on infinitesimal strains. It is proposed that compatibility with the linear theory is an essential feature of any nonlinear model of arterial response. Two approaches towards ensuring such compatibility are proposed. The first is that the nonlinear theory reduces to the full six-constant linear theory, without any restrictions being imposed on the constants. An alternative modelling strategy whereby an anisotropic material is compatible with a simpler material in the linear limit is also proposed. In particular, necessary and sufficient conditions are obtained for a nonlinear anisotropic material to be compatible with an isotropic material for infinitesimal deformations. Materials that satisfy these conditions should be useful in the modelling of the crimped collagen fibres in the undeformed configuration.

The finite compression of elastic solid cylinders in the presence of gravity

1971 ◽  
Vol 321 (1546) ◽  
pp. 381-396 ◽  
Keyword(s):  
Axial Compression ◽  
Energy Function ◽  
Strain Energy ◽  
Body Force ◽  
Strain Energy Function ◽  
Elastic Solid ◽  
Infinitesimal Deformation ◽  
Material Constants ◽  
Gravity Effects ◽  
Theoretical Predictions

The deformation induced by gravity in a solid cylinder is considered. The cylinder is placed on a horizontal frictionless surface with the axis vertical and is treated as an isotropic incompressible elastic material. Further distortion is produced by finite axial compression. Thus the overall deformation is predominantly uniform with a small perturbation superimposed to account for the gravity effects. A particular case is when there is no finite axial compression. The solution then describes the shortening and general infinitesimal deformation associated with the gravitational body force. The results may be used for determining the material constants of soft elastic materials. In particular, the equivalent of Young’s modulus for gelatin has been found by measuring the changes in height of cylindrical specimens when removed from rigid containers and using the appropriate formula derived in the analysis. The results obtained were extremely consistent. In addition, the behaviour of gelatin under finite compression has been examined. By comparing the theoretical predictions with the experimental measurements it is shown that gelatin behaves more as a Mooney material than as a material which has a quadratic form for the associated strain energy function. Values of the two material constants occurring in the Mooney form of the strain energy function are obtained.

scholarly journals A generalized strain approach to anisotropic elasticity

2022 ◽  
Vol 12 (1) ◽  
Author(s):  
M. H. B. M. Shariff
Keyword(s):  
Experimental Data ◽  
Energy Function ◽  
Constitutive Equations ◽  
Strain Energy ◽  
Strain Energy Function ◽  
Single Variable ◽  
Energy Functions ◽  
Anisotropic Strain ◽  
Strain Energy Functions ◽  
Strain Function

AbstractThis work proposes a generalized Lagrangian strain function $$f_\alpha$$ f α (that depends on modified stretches) and a volumetric strain function $$g_\alpha$$ g α (that depends on the determinant of the deformation tensor) to characterize isotropic/anisotropic strain energy functions. With the aid of a spectral approach, the single-variable strain functions enable the development of strain energy functions that are consistent with their infinitesimal counterparts, including the development of a strain energy function for the general anisotropic material that contains the general 4th order classical stiffness tensor. The generality of the single-variable strain functions sets a platform for future development of adequate specific forms of the isotropic/anisotropic strain energy function; future modellers only require to construct specific forms of the functions $$f_\alpha$$ f α and $$g_\alpha$$ g α to model their strain energy functions. The spectral invariants used in the constitutive equation have a clear physical interpretation, which is attractive, in aiding experiment design and the construction of specific forms of the strain energy. Some previous strain energy functions that appeared in the literature can be considered as special cases of the proposed generalized strain energy function. The resulting constitutive equations can be easily converted, to allow the mechanical influence of compressed fibres to be excluded or partial excluded and to model fibre dispersion in collagenous soft tissues. Implementation of the constitutive equations in Finite Element software is discussed. The suggested crude specific strain function forms are able to fit the theory well with experimental data and managed to predict several sets of experimental data.

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