495 ALI et al: A REVIEW AND COMPARISON ON SOME RUBBER ELASTICITY MODELS Journal of Scientific & Industrial Research Vol. 69, July 2010, pp. 495-500 *Author for correspondence Tel: +6 017-2496293; Fax: +6 03-86567122 E-mail: aidy@eng.upm.edu.my A review and comparison on some rubber elasticity models Aidy Ali * , M Hosseini and B B Sahari Department of Mechanical & Manufacturing Engineering, Faculty of Engineering, Universiti Putra Malaysia, UPM, 43400 Serdang, Selangor, Malaysia Received 27 April 2009; revised & accepted 07 April 2010 This study reviews several classical continuum mechanics models for incompressible and isotropic materials based on strain energy potential and then compares ability of neo-Hookean, Yeoh, Mooney-Rivlin and Ogden models in predicting uniaxial deformation states based on experimental data from dumb-bell test specimen under uniaxial loading. Keywords: Curve fitting, Hyperelasticity, Rubber, Strain energy density Introduction Rubber material usually has long chain molecules as polymers. Elastomer is combination of elastic and polymer and is often used interchangeably with rubber 1 . Rubber can withstand very large strains with no permanent deformation or fracture 2 . Elastomers have special physical properties (flexibility, extensibility, resiliency and durability), which are unmatched by other types of materials 3 , however, it still presents behavior in common with other material 4 . This notable characteristics change with fatigue, light, heat, oxygen and ozone, during passing of time 5 . Elastomers present avery complicated mechanical behavior that exceed linear elastic theory and contain large deformations, plastic and viscoelastic properties and stress softening 6,7 . This characteristic presents complications to modeling of elastomers compared with other traditional engineering materials 8 . Under three physical states of a polymer 9,10 , a glassy polymer is brittle. A crystalline polymer pass sequence of changes consist of, elastic deformation, yield, plastic flow, necking, strain hardening and strain fracture. Rubbers are unique in being soft, very extensible and very elastic. This study reviews and compares recent models, which are offered in hyperelastic materials and discuss their abilities in predicting uniaxial deformation states based on experimental data from dumb-bell test specimen under uniaxial loading. Elasticity By a simple assumption of linear stress-strain relationship, rubber can be considered as a linearly elastic material at small strains. However, for analyzing rubber behavior in large deformation, large elastic deformation theory should be considered 11 . Elastic Properties at Large Deformations According to Rivlin’s phenomenological theory, rubber is assumed isotropic in elastic behavior and very nearly incompressible. Elastic properties of rubber can be explained as a strain energy function based on strain invariants (I 1 , I 2 and I 3 ). Stress and strain analysis problems may be solved independent of microscopic system or molecular concepts and elasticity theory can be starting point of any kind of modeling effort as 12-14 ( ) 3 2 1 , , I I I f W = ...(1) where W (or U) is strain energy density, and I 1 , I 2 and I 3 are invariants of Green deformation tensor given in terms of principle extension ratios as 2 3 2 2 2 1 1 λ λ λ + + = I , 2 1 2 3 2 3 2 2 2 2 2 1 2 λ λ λ λ λ λ + + = I and 2 3 2 2 2 1 3 λ λ λ = I . Eq. (1) can be given as ( ) ( ) ( ) k j i k j i ijk I I I C W 1 3 3 3 2 1 1 - ⋅ - ⋅ - = ∑ ∞ = + + ...(2) A simple extension is defined by λ λ = 1 , 2 1 3 2 - = = λ λ λ . Incompressibility condition is