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.

Large deformation isotropic elasticity: on the correlation of theory and experiment for compressible rubberlike solids

1972 ◽  
Vol 328 (1575) ◽  
pp. 567-583 ◽  
Keyword(s):  
Energy Function ◽  
Strain Energy ◽  
Pure Shear ◽  
Density Ratio ◽  
Strain Energy Function ◽  
Elastic Solids ◽  
Simple Tension ◽  
Isotropic Elasticity ◽  
Full Discussion ◽  
Theory And Experiment

A method of approach to the correlation of theory and experiment for incompressible isotropic elastic solids under finite strain was developed in a previous paper (Ogden 1972). Here, the results of that work are extended to incorporate the effects of compressibility (under isothermal conditions). The strain-energy function constructed for incompressible materials is augmented by a function of the density ratio with the result that experimental data on the compressibility of rubberlike materials are adequately accounted for. At the same time the good fit of the strain-energy function arising in the incompressibility theory to the data in simple tension, pure shear and equibiaxial tension is maintained in the compressible theory without any change in the values of the material constants. A full discussion of inequalities which may reasonably be imposed upon the material parameters occurring in the compressible theory is included.

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.

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.

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.

On H. Hencky’s Approximate Strain-Energy Function for Moderate Deformations

1979 ◽  
Vol 46 (1) ◽  
pp. 78-82 ◽  
Author(s):  
L. Anand
Keyword(s):  
Energy Function ◽  
Strain Energy ◽  
Finite Strain ◽  
Large Deformations ◽  
Strain Energy Function ◽  
Strain Measure ◽  
Logarithmic Measure ◽  
Infinitesimal Strain ◽  
Isotropic Elasticity ◽  
Good Agreement

It is shown that the classical strain-energy function of infinitesimal isotropic elasticity is in good agreement with experiment for a wide class of materials for moderately large deformations, provided the infinitesimal strain measure occurring in the strain-energy function is replaced by the Hencky or logarithmic measure of finite strain.

scholarly journals A Bimodular Polyconvex Anisotropic Strain Energy Function for Articular Cartilage

2006 ◽  
Vol 129 (2) ◽  
pp. 250-258 ◽  
Author(s):  
Stephen M. Klisch
Keyword(s):  
Articular Cartilage ◽  
Strong Interaction ◽  
Energy Function ◽  
Strain Energy ◽  
Mechanical Response ◽  
Deformation Gradient ◽  
Strain Energy Function ◽  
Finite Deformations ◽  
Orthotropic Materials ◽  
Interaction Terms

A strain energy function for finite deformations is developed that has the capability to describe the nonlinear, anisotropic, and asymmetric mechanical response that is typical of articular cartilage. In particular, the bimodular feature is employed by including strain energy terms that are only mechanically active when the corresponding fiber directions are in tension. Furthermore, the strain energy function is a polyconvex function of the deformation gradient tensor so that it meets material stability criteria. A novel feature of the model is the use of bimodular and polyconvex “strong interaction terms” for the strain invariants of orthotropic materials. Several regression analyses are performed using a hypothetical experimental dataset that captures the anisotropic and asymmetric behavior of articular cartilage. The results suggest that the main advantage of a model employing the strong interaction terms is to provide the capability for modeling anisotropic and asymmetric Poisson’s ratios, as well as axial stress–axial strain responses, in tension and compression for finite deformations.

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.

Application of Finite Elastic Theory to the Behavior of Rubberlike Materials

1963 ◽  
Vol 36 (5) ◽  
pp. 1459-1496 ◽  
Author(s):  
Paul J. Blatz
Keyword(s):  
Energy Function ◽  
Strain Energy ◽  
Mechanical Response ◽  
Finite Elasticity ◽  
Strain Energy Function ◽  
Elastic Theory ◽  
Mechanical Parameters ◽  
Molecular Interpretation ◽  
Rubberlike Materials

Abstract A brief review of the theory of finite elasticity is presented. The theory is applied to the characterization of the mechanical response parameters of a polyurethan foam. The incorporation of compressibility and anisotropy effects into the strain energy function are discussed. An example of the behavior of a composite or filled foam is presented. Finally some of the problems associated with the molecular interpretation of mechanical parameters are discussed.

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.

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.

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