Rheology of brain tissue and hydrogels: A novel hyperelastic and viscoelastic model for forensic applications

A new strain-energy function W, which possesses the strain energy expressible as a rational function of the principal invariants of the Cauchy tensor C, is proposed. It generates a hyperelastic constitutive equation with characteristics of brain tissues: a much stronger resistance to compression than to stretching and strongly nonlinear response in simple shear, including non-zero first and second normal differences. This model exponent α resembles the Ogden model in uniaxial stretching/compression and reveals plausible predictions for brain tissue with even values of α < 0 with sufficiently high magnitude (say, at α = −20). However, the dependence of the strain-energy function W on the principal invariants of C links it to hyperelastic hydrogel models (the Special and General Blatz–Ko models, neo-Hookean materials, incompressible Mooney–Rivlin and the Yeoh models). For α = −8, the present model reveals a compression/stretching behavior close to the tensorial Special Blatz–Ko model used for description of hydrogels. Furthermore, the present hyperelastic model is used as a kernel of the corresponding tensorial viscoelastic model with exponential fading memory. It belongs to the class of the integral Bernstein–Kearsley–Zapas (BKZ) models. In a number of important cases (the uniaxial stretching/compression, simple shear), it can be transformed into a differential viscoelastic model and predict viscoelastic liquid-like behavior under sustained deformations. The stress relaxation following an imposed strain reduces to the hyperelastic model with the elastic parameters exponentially fading in time. These tensorial hyperelastic and viscoelastic constitutive equations aim applications in modeling of blast-induced traumatic brain injuries and bullet penetration and spatter of brain tissue in forensic context.