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.

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.

A Simple Transversely Isotropic Hyperelastic Constitutive Model Suitable for Finite Element Analysis of Fiber Reinforced Elastomers

2011 ◽  
Vol 133 (2) ◽  
Author(s):  
Leslee W. Brown ◽  
Lorenzo M. Smith
Keyword(s):  
Finite Element Analysis ◽  
Finite Element ◽  
Energy Function ◽  
Strain Energy ◽  
Strain Energy Function ◽  
Transversely Isotropic ◽  
Material Model ◽  
Fiber Reinforced ◽  
Element Analysis ◽  
Material Tests

A transversely isotropic fiber reinforced elastomer’s hyperelasticity is characterized using a series of constitutive tests (uniaxial tension, uniaxial compression, simple shear, and constrained compression test). A suitable transversely isotropic hyperelastic invariant based strain energy function is proposed and methods for determining the material coefficients are shown. This material model is implemented in a finite element analysis by creating a user subroutine for a commercial finite element code and then used to analyze the material tests. A useful set of constitutive material data for multiple modes of deformation is given. The proposed strain energy function fits the experimental data reasonably well over the strain region of interest. Finite element analysis of the material tests reveals further insight into the materials constitutive nature. The proposed strain energy function is suitable for finite element use by the practicing engineer for small to moderate strains. The necessary material coefficients can be determined from a few simple laboratory tests.

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.

Nonlinear Finite Element Analysis of Crack Tip of Rubber-Like Material

2007 ◽  
Vol 353-358 ◽  
pp. 1013-1016
Author(s):  
Jian Bing Sang ◽  
Su Fang Xing ◽  
Xiao Lei Li ◽  
Jie Zhang
Keyword(s):  
Finite Element ◽  
Energy Function ◽  
Strain Energy ◽  
Strain Energy Function ◽  
Elastic Behavior ◽  
Element Analysis ◽  
Finite Element Program ◽  
User Subroutine ◽  
Element Program ◽  
Notch Tip

It has been well known that rubber-like material can undergo large deformation and exhibit large nonlinear elastic behavior. Because of the geometrically nonlinear of rubber like material, it is more difficult to analyze it with finite element near the notch tip. What is more, because there are varieties of the strain energy functions, implementation of these models in a general finite element program to meet the need of industry applications can be time consuming. In order to make use of the constitutive equation of Y.C. Gao in 1997 and analyze the notch tip of rubber-like material, a framework to implement the rubber-like material model is established within the general-purpose finite element program MSC.Marc. It will be very convenient to implement this isotropic hyperelastic model into the program with a user subroutine. This paper starts with the theoretical analysis based on the strain energy function given by Y.C. Gao in 1997. A user subroutine is programmed to implement this strain energy function into the program of MSC.Marc, which offer a convenient method to analyze the stress and strain of rubber-like material with the strain energy function that is needed. Though analysis with MSC.Marc, it is found that the result with finite element is consistent with the analytical result that given by Y.C. Gao in 1997, which testify that analyzing rubber like material with this method is reasonable and convenient.

On the Ogden Strain-Energy Function

1997 ◽  
Vol 70 (2) ◽  
pp. 175-182 ◽  
Author(s):  
O. H. Yeoh
Keyword(s):  
Regression Analysis ◽  
Energy Function ◽  
Strain Energy ◽  
Strain Energy Function ◽  
Material Model ◽  
Nonlinear Regression Analysis ◽  
Element Analysis ◽  
Strain Data ◽  
Predicting Behavior ◽  
Deformation Modes

Abstract The Ogden strain-energy function for rubber appears to be gaining popularity among users of finite element analysis. This paper discusses some of the special features of this material model. It explains why nonlinear regression analysis of stress-strain data obtained from just one mode of deformation may yield an Ogden strain-energy function that is unsatisfactory for predicting behavior in other deformation modes. It suggests the regression analysis be constrained such that some of the coefficients are chosen based upon the known behavior of rubbery materials.

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.

The Elasticity of Rubber

1958 ◽  
Vol 31 (5) ◽  
pp. 959-981
Author(s):  
Lawrence A. Wood
Keyword(s):  
Energy Function ◽  
Strain Energy ◽  
Strain Energy Function ◽  
Phenomenological Approach ◽  
Entropy Changes ◽  
A Value ◽  
Strain Invariants ◽  
Strain Invariant ◽  
The One ◽  
The Relationship

Abstract Rubbers, natural and synthetic, are unique in being highly extensible and in retracting forcibly and quickly to substantially their original dimensions when released. It has been found that the stress-strain curves for extension and compression of most of the simplest vulcanizates of natural rubber and the three most important synthetic rubbers are similar in shape. The relationship is expressed by the equation F/M=(L−1−L−2) exp A(L−L−1) where F is the stress, L the ratio of stretched to unstretched length, and M and A are constants. The constant M depends on the nature of the rubber, the extent of vulcanization, and the time of creep. The constant A has a value of about 0.38. By a study of stress-temperature relations it is found that the most important factor in the retraction of stretched rubber is the tendency of long chain flexible molecules to return to a configuration which is statistically more probable than the one which the stretching has forced them to assume. Calculations of entropy changes arising from stretching can be made from probability considerations, and a strain energy function deduced from the entropy changes. Stresses calculated from the strain energy function agree with those observed in compression but are greater than those observed in extension by almost 50 per cent at L=3. A phenomenological approach shows that the strain energy should be expressible as a function of certain quantities called strain invariants, calculable from the deformations. The simplest behavior is found in the region of compression (L less than 1), where the strain energy is merely the first invariant times a constant calculable from the entropy changes. For values of L between 1.5 and 3 a different constant and an added term involving the second strain invariant are required. The explanation of this behavior in molecular terms is one of the most important current problems of rubber elasticity.

Constitutive Equations for Isotropic Rubber-Like Materials Using Phenomenological Approach: A Bibliography (1930–2003)

2006 ◽  
Vol 79 (3) ◽  
pp. 489-499 ◽  
Author(s):  
Vahap Vahapoğlu ◽  
Sami Karadeniz
Keyword(s):  
Energy Function ◽  
Constitutive Equations ◽  
Strain Energy ◽  
Isothermal Condition ◽  
Strain Energy Function ◽  
Phenomenological Approach ◽  
Elastic Behavior ◽  
Energy Functions ◽  
Strain Energy Functions

Abstract To describe the elastic behavior of rubber-like materials, numerous specific forms of strain energy functions have been proposed in the literature. This bibliography provides a list of references on the strain energy functions for rubber-like materials on isothermal condition using the phenomenological approach. The published works, either containing the strain energy function proposals or the discussions on such proposals, based upon the phenomenological approach, are classified.

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