A hyperelastic constitutive model for rubber-like materials H. Khajehsaeid a , J. Arghavani a , R. Naghdabadi a, b, * a Department of Mechanical Engineering, Sharif University of Technology, Tehran, Iran b Institute for Nano-Science and Technology, Sharif University of Technology, Tehran, Iran article info Article history: Received 15 March 2012 Accepted 23 September 2012 Available online 17 October 2012 Keywords: Hyperelasticity Rubber-like materials Strain energy function abstract Hyperelastic behavior of isotropic incompressible rubbers is studied to develop a strain energy function which satisfies all the necessary characteristic properties of an efficient hyperelastic model. The proposed strain energy function includes only three material parameters which are somehow related to the physical quantities of the material molecular network. Moreover, the model benefits from mathematical simplicity, well suitting in all ranges of stretch and possessing the property of deformation-mode-independency. This reduces the required number of experimental tests for parameter calibration of the model. Results of the proposed model are compared with results of some available models as well as experimental data. Moreover, complete analysis of the Mooney plot over a wide range of stretch in extensionecompression is carried out. It is found that the proposed model gives reasonable predictions in comparison with those of experiments. Ó 2012 Elsevier Masson SAS. All rights reserved. 1. Introduction Rubber-like materials consist of randomly oriented long molecular chains with cross-linking at junctions called points in a network configuration (Fig. 1). The term elastomer (combination of elastic and polymer) is also used instead of rubber (Smith et al., 1993). The most outstanding property of elastomers is their ability to undergo large deformations under relatively small stresses and to retain initial configuration without considerable permanent deformation after stress removal. This behavior is mostly governed by changes of network entropy as the orientation of chains alters with deformation. These materials have been used in numerous applications such as structural bearings, tires, medical devices and vibration isolators due to their peculiar properties (Brinson and Brinson, 2008). Elastomers exhibit a complicated nonlinear behavior including hysteresis, viscoelasticity and stress softening phenomenon. The latter case called Mullins effect, occurs in first cycles of loading and have been studied extensively (Bueche, 1960; Marckmann et al., 2002; Chagnon et al., 2006; Diani et al., 2006). Hysteresis and viscoelastic behavior take place due to deviation from static deformation, in which the effects of time and deformation rate must be taken into account (Bergstrom and Boyce, 1998; Reese, 2003; Amin et al., 2006; Li and Lua, 2009). In the case of static deformations, rubbers exhibit hyperelastic behavior; Thus, a strain energy density function, W, can be attributed. We highlight that, the present work focuses on the hyperelastic behavior, knowing that, other phenomena can be considered in the proposed model by means of appropriate modifications. For isotropic materials (e.g., rubbers), strain energy function (SEF) can be represented in terms of right (or left) CauchyeGreen tensor C (or B) invariants (I 1 , I 2 , I 3 ) or eigen values of deformation gradient tensor F, called principal stretches (l 1 , l 2 , l 3 ). i.e., W ¼ WðI 1 ; I 2 ; I 3 Þ or W ¼ Wðl 1 ; l 2 ; l 3 Þ (1) where I 1 ¼ trðCÞ I 2 ¼ 1 2 h I 2 1  tr  C 2 i I 3 ¼ detðCÞ (2) tr and det demonstrate trace and determinant of a tensor. The form of strain energy density function depends on symmetry, thermo- dynamic, energetic and entropic considerations. For example, in the case of rubber-like materials, due to the negligible compressibility under conventional pressures, incompressibility assumption is often used (I 3 ¼ 1). Therefore, we conclude that: W ¼ WðI 1 ; I 2 Þ (3) * Corresponding author. Department of Mechanical Engineering, Sharif Univer- sity of Technology, Tehran, Iran. Tel.: þ98 21 6616 5546; fax: þ98 21 6600 0021. E-mail address: naghdabd@sharif.edu (R. Naghdabadi). Contents lists available at SciVerse ScienceDirect European Journal of Mechanics A/Solids journal homepage: www.elsevier.com/locate/ejmsol 0997-7538/$ e see front matter Ó 2012 Elsevier Masson SAS. All rights reserved. http://dx.doi.org/10.1016/j.euromechsol.2012.09.010 European Journal of Mechanics A/Solids 38 (2013) 144e151