A Hyperelastic Constitutive Law for Aortic Valve Tissue

2009 ◽  
Vol 131 (8) ◽  
Author(s):  
Karen May-Newman ◽  
Charles Lam ◽  
Frank C. P. Yin
Keyword(s):  
Aortic Valve ◽  
Energy Function ◽  
Strain Energy ◽  
Strain Energy Function ◽  
Material Behavior ◽  
Constitutive Law ◽  
Biaxial Testing ◽  
Strain Behavior ◽  
Wide Range ◽  
Strain Invariants

The objective of the present study was to perform biaxial testing and apply constitutive modeling to develop a strain energy function that accurately predicts the material behavior of the aortic valve leaflets. Ten leaflets from seven normal porcine aortic valves were biaxially stretched in a variety of protocols and the data combined to develop and fit a strain energy function to describe the material behavior. The results showed that the nonlinear anisotropic behavior of the aortic valve is well described by a strain energy function of two strain invariants, which uses only three coefficients to accurately predict the stress-strain behavior over a wide range of deformations. This structurally-motivated constitutive law has many applications, including computational modeling for clinical and engineering valve treatments.

Some Forms of the Strain Energy Function for Rubber

1993 ◽  
Vol 66 (5) ◽  
pp. 754-771 ◽  
Author(s):  
O. H. Yeoh
Keyword(s):  
Energy Function ◽  
Strain Energy ◽  
Large Strain ◽  
Strain Energy Function ◽  
Phenomenological Theory ◽  
Material Model ◽  
Elastic Behavior ◽  
Strain Behavior ◽  
Strain Invariants ◽  
Infinite Power

Abstract According to Rivlin's Phenomenological Theory of Rubber Elasticity, the elastic properties of a rubber may be described in terms of a strain energy function which is an infinite power series in the strain invariants I1, I2 and I3. The simplest forms of Rivlin's strain energy function are the neo-Hookean, which is obtained by truncating the infinite series to just the first term in I1, and the Mooney-Rivlin, which retains the first terms in I1 and I2. Recently, we proposed a strain energy function which is a cubic in I1. Conceptually, the proposed function is a material model with a shear modulus that varies with deformation. In this paper, we compare the large strain behavior of rubber as predicted by these forms of the strain energy function. The elastic behavior of swollen rubber is also discussed.

Experimentally Tractable, Pseudo-elastic Constitutive Law for Biomembranes: I. Theory

2003 ◽  
Vol 125 (1) ◽  
pp. 94-99 ◽  
Author(s):  
John C. Criscione ◽  
Michael S. Sacks ◽  
William C. Hunter
Keyword(s):  
Stress Response ◽  
Energy Function ◽  
Strain Energy ◽  
Strain Energy Function ◽  
Scalar Function ◽  
Constitutive Law ◽  
Biaxial Testing ◽  
Reference Volume ◽  
Stress Strain Relation ◽  
Work Done

Although visco-elastic in general, the stress-strain relation of biomembranes is one-to-one or pseudo-elastic when being loaded after preconditioning. This pseudo-elastic relation is hypoelastic (i.e., it is not hyperelastic), yet much of the stress response can be characterized by a scalar function Ω that represents the work done (per unit reference volume) on the specimen during loading. (Since a pseudo-strain-energy function W is optimized to fit the test data and not the work done, Ω is not equal to W in general.) The remaining part tR of the stress response does no work during loading. With biaxial testing, Ω can be definitively determined from data. Moreover, for tests with the stretch directions coaxial to the axes of anisotropy, tR can be accurately characterized by a scalar function ω that depends on the strain. This paper is part 1 of 2 with “I. Theory” and “II. Application.”

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.

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.

A note on the finite plane-strain equations for isotropic incompressible materials

1955 ◽  
Vol 51 (2) ◽  
pp. 363-367 ◽  
Author(s):  
J. E. Adkins
Keyword(s):  
Differential Equations ◽  
Plane Strain ◽  
Energy Function ◽  
Strain Energy ◽  
Strain Energy Function ◽  
Theory Of Elasticity ◽  
Finite Plane ◽  
Incompressible Materials ◽  
Stress Components ◽  
Strain Invariants

For elastic deformations beyond the range of the classical infinitesimal theory of elasticity, the governing differential equations are non-linear in form, and orthodox methods of solution are not usually applicable. Simplifying features appear, however, when a restriction is imposed either upon the form of the deformation, or upon the form of strain-energy function employed to define the elastic properties of the material. Thus in the problems of torsion and flexure considered by Rivlin (4, 5, 6) it is possible to avoid introducing partial differential equations into the analysis, while in the theory of finite plane strain developed by Adkins, Green and Shield (1) the reduction in the number of dependent and independent variables involved introduces some measure of simplicity. Some further simplification is achieved when the strain-energy function can be considered as a linear function of the strain invariants as postulated by Mooney(2) for incompressible materials. In the present paper the plane-strain equations for a Mooney material are reduced to symmetrical forms which do not involve the stress components, and some special solutions of these equations are derived.

A Structural Strain Energy Function for Vascular Tissue Considering Elastin Anisotropy

2007 ◽  
Author(s):  
Rana Rezakhaniha ◽  
Nikos Stergiopulos
Keyword(s):  
Energy Function ◽  
Constitutive Equations ◽  
Vessel Wall ◽  
Strain Energy ◽  
Vascular Tissue ◽  
Strain Energy Function ◽  
Nonlinear Properties ◽  
Wide Range ◽  
Nonlinear Elastic ◽  
Structural Strain

The vessel wall exhibits relatively strong nonlinear properties and undergoes a wide range of deformations. Identification of a strain energy function (SEF) is the preferred method to describe the complex nonlinear elastic properties of the vascular tissue. Once the strain energy function is known, constitutive equations can be obtained.

scholarly journals The deformation of rubber cylinders and tubes by rotation

1977 ◽  
Vol 20 (1) ◽  
pp. 62-96 ◽  
Author(s):  
P. Chadwick ◽  
C. F. M. Creasy ◽  
V. G. Hart
Keyword(s):  
Energy Function ◽  
Strain Energy ◽  
Numerical Study ◽  
Strain Energy Function ◽  
Circular Cylinders ◽  
Cylindrical Shape ◽  
Vulcanized Rubber ◽  
Wide Range ◽  
Steady Rotation ◽  
Axis Of Symmetry

AbstractA detailed analytical and numerical study is made of the deformation of highly elastic circular cylinders and tubes produced by steady rotation about the axis of symmetry. Explicit results are obtained through the use of Ogden's strain–energy function for incompressible isotropic elastic materials which, as well as being analytically convenient, is capable of reproducing accurately the observed isothermal behaviour of vulcanized rubber over a wide range of deformations. The three problems of steady rotation considered here concern (i) a tube shrink-fitted to a rigid spindle, (ii)an unconstrained tube, and (iii) a solid cylinder. In each case a set of restictions on the material constans appearing in the strain–energy function is stated which ensures that a tubular of cylindrical shape-preserving deformation exists for all angular spees and that, for problems (i) and (iii), there is no other solution. In connection with problems (ii) and (iii) values of the material constans are also given which correspond to the bifuraction or non-existence of soultions. Enegry consideration are used to determine the local stability of the various solutions obtained.

Nonlinear Thermohyperviscoelastic Constitutive Model for Soft Materials with Strain Rate and Temperature Dependency

2020 ◽  
Vol 12 (06) ◽  
pp. 2050059
Author(s):  
Zahra Matin Ghahfarokhi ◽  
Mehdi Salmani-Tehrani ◽  
Mahdi Moghimi Zand
Keyword(s):  
Constitutive Model ◽  
Strain Rate ◽  
Energy Function ◽  
Strain Energy ◽  
Strain Energy Function ◽  
Soft Materials ◽  
Material Behavior ◽  
Temperature Dependent ◽  
Dependent Behavior ◽  
Temperature Dependent Behavior

Soft materials, such as polymeric materials and biological tissues, often exhibit strain rate and temperature-dependent behavior when subjected to external loads. To characterize the thermomechanical behavior of isotropic soft material, a thermohyperviscoelastic constitutive model has been developed through an additive decomposition of strain energy function into elastic and viscous parts. A three-term generalized Rivlin strain energy function is utilized to formulate the hyperelastic part of the model, while a new viscous potential function is proposed to describe the effect of strain rate and temperature on material behavior. Toward this end, a new procedure has been proposed to determine the viscous mechanical properties as a function of strain-rate and temperature. Comparing with the previously published experimental data for linear low-density polyethylene reveals that the proposed model can sufficiently capture the nonlinearity, rate- and temperature-dependent behavior of the soft materials.

Some general results in the theory of large elastic deformations

1955 ◽  
Vol 231 (1184) ◽  
pp. 75-90 ◽  
Keyword(s):  
Elastic Body ◽  
Energy Function ◽  
Strain Energy ◽  
General Type ◽  
Cylindrical Tube ◽  
Strain Energy Function ◽  
General Function ◽  
Cylindrical Annulus ◽  
Special Cases ◽  
Wide Range

When the strain-energy function for an elastic body is expressed as a function of the six components of strain, the solution of a given problem for different types of material may assume very different forms. In the present paper, by regarding the strain-energy function as a function of the parameters defining the deformation, results are obtained which are valid for a wide range of materials. The analysis for each problem is performed initially for bodies possessing a suitable type of curvilinear aeolotropy, and results are derived which are independent of symmetries in the elastic material. These results are therefore valid, not only for the general type of material initially considered, but also for isotropic bodies and for materials which are orthotropic or transversely isotropic with respect to the curvilinear co-ordinate system which defines the aeolotropy. Both compressible and incompressible bodies are considered. From this point of view, a general type of cylindrically symmetrical deformation is examined which includes as special cases the problem of flexure, the inflation, extension and torsion of a cylindrical tube, and the shear of a cylindrical annulus. Particular results for these special cases are considered separately, and for the flexure and torsion problems, expressions are found for the resultant forces and couples required to maintain the deformation. A brief analysis is also given for the corresponding types of deformation for a cuboid. In the final section of the paper, a generalized shear problem is considered in which, during deformation, each point of the elastic body moves parallel to a given axis through a distance which is a general function of position in a plane normal to that axis.

A Microstructurally Based Orthotropic Hyperelastic Constitutive Law

2002 ◽  
Vol 69 (5) ◽  
pp. 570-579 ◽  
Author(s):  
J. E. Bischoff ◽  
E. A. Arruda ◽  
K. Grosh
Keyword(s):  
Finite Element Method ◽  
Finite Element ◽  
Constitutive Model ◽  
Energy Function ◽  
Strain Energy ◽  
Strain Energy Function ◽  
Entropy Change ◽  
Constitutive Law ◽  
Elastic Behavior ◽  
The Finite Element Method

A constitutive model is developed to characterize a general class of polymer and polymer-like materials that displays hyperelastic orthotropic mechanical behavior. The strain energy function is derived from the entropy change associated with the deformation of constituent macromolecules and the strain energy change associated with the deformation of a representative orthotropic unit cell. The ability of this model to predict nonlinear, orthotropic elastic behavior is examined by comparing the theory to experimental results in the literature. Simulations of more complicated boundary value problems are performed using the finite element method.

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