A constitutive description of elastomer behaviour at high strain rates e A strain- dependent relaxation time approach H. Pouriayevali * , Y.B. Guo, V.P.W. Shim Impact Mechanics Laboratory, Department of Mechanical Engineering, National University of Singapore, 9 Engineering Drive 1, Singapore 117576, Singapore article info Article history: Received 20 December 2011 Received in revised form 1 April 2012 Accepted 3 April 2012 Available online 7 April 2012 Keywords: Relaxation time High strain rate Rubber Visco-hyperelasticity abstract A strain-dependent relaxation time perspective is proposed for a visco-hyperelastic constitutive equation to describe the large compressive and tensile deformation response of incompressible elastomeric materials at high strain rates. The description comprises two components: the first characterizes quasi- static nonlinear response corresponding to hyperelasticity, using a polynomial strain energy density function, while the second is an integral form of first and captures rate sensitivity by incorporating a deformation-dependent relaxation time function. The model is applied to describe the response of six types of elastomer with different hardnesses, namely U50, U70 polyurethane rubber, SHA40, 60, 80 rubber, and Ethylene-Propylene-Diene-Monomer (EPDM) rubber. Material samples are subjected to quasi-static and dynamic loading using a universal testing machine and a Split Hopkinson Bar device respectively. The proposed equation is able to track the experimental responses and demonstrate the potential to predict the dynamic behaviour of elastomeric material over a range of strain rates. Ó 2012 Elsevier Ltd. All rights reserved. 1. Introduction Elastomers are able to accommodate large deformations and possess damping characteristics, making them suitable for employment in the dissipation of kinetic energy associated with impacts and shocks. Therefore, analysis and modelling of the dynamic response of elastomers over a wide range of strains and strain rates are essential and will facilitate the use of computer simulation for designing products that incorporate elastomeric padding and components. The stressestrain responses of elastomers generally exhibit nonlinear rate-dependent elastic behaviour associated with negli- gible residual strain after unloading from a large deformation [1,2]. The observed rate-dependence corresponds to the readjustment of molecular chains, whereby the applied load is accommodated through various relaxation processes (e.g. rearrangement, reor- ientation, uncoiling, etc., of chains). An elastomer contains a wide range of molecular chain lengths - e.g. in localized regions with relatively short chains, as well as long convolutions that span larger areas. These chains are cross-linked and spiral and entangle among themselves or with neighbours. They are also able to slip and rearrange to accommodate relaxation [3e5]. Tosaka et al. [6] suggested that when relatively small strains are applied to elastomers, the deformation is accommodated primarily within localized regions; deformation extends throughout the material as the strain is increased. Relaxation processes occur relatively rapidly within localized regions containing primarily shorter chains, while rearrangements of long convolutions require greater relaxation times [7]. Therefore, relaxation processes asso- ciated with smaller relaxation times, are activated at smaller strains, while relaxation for larger strains is linked to processes including long convolutions and greater relaxation times. Conse- quently, it is expected that the relaxation time is influenced by the degree of strain applied, because this determines which compo- nents of deformation corresponding to the various molecular chain length, are dominant. When high rate deformation is applied, the material does not have sufficient time for all relaxation processes to be completed, and the material response is thus affected by incomplete rear- rangement of chains. The relationship between the strain experi- enced and relaxation time is therefore investigated in this study. Nonlinear-viscoelastic constitutive models have been proposed and developed to predict the behaviour of rubber-like materials [8e10]. These models can be classified according to two approaches: differential equations derived from micromechanics modelling [11e 13], and history-integral models based on macromechanical theories associated with a fading-memory effect [14,15]. Arruda and Boyce [11] developed a rate-independent three-dimensional constitutive relationship based on an eight-chain representation of * Corresponding author. Tel.: þ65 6516 2228; fax: þ65 6779 1459. E-mail address: pouriayevali.habib@nus.edu.sg (H. Pouriayevali). Contents lists available at SciVerse ScienceDirect International Journal of Impact Engineering journal homepage: www.elsevier.com/locate/ijimpeng 0734-743X/$ e see front matter Ó 2012 Elsevier Ltd. All rights reserved. doi:10.1016/j.ijimpeng.2012.04.001 International Journal of Impact Engineering 47 (2012) 71e78