Constitutive model for brain tissue under finite compression Kaveh Laksari, Mehdi Shafieian, Kurosh Darvish n Department of Mechanical Engineering, Temple University, 1947N. 12th Street, Philadelphia, PA 19122, USA article info Article history: Accepted 19 December 2011 Keywords: Brain tissue biomechanics Finite deformation Viscoelasticity Compression test Isochronous curves abstract While advances in computational models of mechanical phenomena have made it possible to simulate dynamically complex problems in biomechanics, accurate material models for soft tissues, particularly brain tissue, have proven to be very challenging. Most studies in the literature on material properties of brain tissue are performed in shear loading and very few tackle the behavior of brain in compression. In this study, a viscoelastic constitutive model of bovine brain tissue under finite step-and-hold uniaxial compression with 10 s –1 ramp rate and 20 s hold time has been developed. The assumption of quasi- linear viscoelasticity (QLV) was validated for strain levels of up to 35%. A generalized Rivlin model was used for the isochoric part of the deformation and it was shown that at least three terms (C 10 , C 01 and C 11 ) are needed to accurately capture the material behavior. Furthermore, for the volumetric deformation, a two parameter Ogden model was used and the extent of material incompressibility was studied. The hyperelastic material parameters were determined through extracting and fitting to two isochronous curves (0.06 s and 14 s) approximating the instantaneous and steady-state elastic responses. Viscoelastic relaxation was characterized at five decay rates (100, 10, 1, 0.1, 0 s 1 ) and the results in compression and their extrapolation to tension were compared against previous models. & 2012 Elsevier Ltd. All rights reserved. 1. Introduction Characterizing mechanical properties of soft tissues and in particular brain tissue is a major concern in biomechanics, especially with the reported number of Traumatic Brain Injuries (TBI) that at around 52,000 deaths each year ranks as the most significant cause of fatal injuries (CDC, 2011). Recently with the advances in finite element modeling of brain tissue under impact loading, the issue of material properties has received greater attention. Studies on deformation of brain tissue in closed head impact experiments have shown that in order to have realistic models of such phenomena, theory of large deformation needs to be utilized, i.e., models based on the infinitesimal theory of continuum mechanics and linear viscoelasticity will not give accurate results. Various techniques such as stress-relaxation, creep, and oscil- lation tests have been used to determine the needed material models in shear, compression and tension modes. Among such experiments, few results have been reported for compression of brain tissue especially at higher rates required to characterize impact related injuries. The goal of this study was to model the material properties of brain tissue under uniaxial compression and develop a nonlinear viscoelastic constitutive model useful for dynamic loading simulations. Estes and McElhaney (1970) conducted uniaxial compression experiments on human and Rhesus monkey brain tissues at strain rates varying between 0.08 and 40 s 1 . They observed that during the loading ramp, the stress curves are all concave upward, contain no linear portion, and become stiffer as the rate of the applied strain increases. Mendis et al. (1995) developed a 2-term Mooney–Rivlin hyperelastic model for the steady-state response of brain tissue samples using experiments performed by Estes and McElhaney (1970). Based on the values of the reduced stress function derived from the loading section of the stress histories, they determined the material parameters, giving the steady-state shear modulus as 2748 Pa. Their model exhibits significant non- linearity in compression but linear behavior in shear. Miller and Chinzei (1997) performed experiments on porcine brain samples to develop a constitutive model for brain tissue under unconfined uniaxial compression and used a nonlinear viscoelastic model to describe the material behavior. They employed two additional terms compared to Mendis et al. (1995) and used three strain rates to determine the material parameters for equilibrium and characteristic time constants of 50 s and 0.5 s and the corresponding linear shear moduli reported were 1080 Pa, 214 Pa, and 162 Pa, respectively. Miller and Chinzei (2002) also performed uniaxial tensile experiments on brain tissue with the specimens glued at the top and bottom plates. In order to compensate for the boundary Contents lists available at SciVerse ScienceDirect journal homepage: www.elsevier.com/locate/jbiomech www.JBiomech.com Journal of Biomechanics 0021-9290/$ - see front matter & 2012 Elsevier Ltd. All rights reserved. doi:10.1016/j.jbiomech.2011.12.023 n Corresponding Author. Tel.: þ215 204 4307; fax: þ215 204 4956. E-mail address: kdarvish@temple.edu (K. Darvish). Please cite this article as: Laksari, K., et al., Constitutive model for brain tissue under finite compression. Journal of Biomechanics (2012), doi:10.1016/j.jbiomech.2011.12.023 Journal of Biomechanics ] (]]]]) ]]]–]]]