Head Injury Simulation in Road Traffic Accidents by Fábio A. O. Fernandes Ricardo J. Alves de Sousa & Mariusz Ptak

Author:Fábio A. O. Fernandes, Ricardo J. Alves de Sousa & Mariusz Ptak
Language: eng
Format: epub
Publisher: Springer International Publishing, Cham

(2.3)

where are the deviatoric principal stretches, which can be obtained through the relation between the total volume ratio J and the the principal stretches ,

(2.4)

N, , and are material parameters. The initial shear modulus can be obtained through:

(2.5)

The bulk modulus for the Ogden form is given by:

(2.6)

Thus, the one-term Ogden hyperelastic function is given by:

(2.7)

If a material is incompressible, the third strain invariant has a value of 1, and the strain energy function is only a function of the first two invariants. Thus, an isotropic hyperelastic incompressible material is characterised by a strain-energy density function W, which is a function of two principal strain invariants only. The stress-strain relationship is then obtained from a partial derivative of the strain energy potential with respect to deformation gradient tensor .

The elastic and viscoelastic behaviour of brain tissue can be characterised using an Ogden based nonlinear viscoelastic model (Rashid et al. 2012a). The relaxation response is based on a Prony series and the strain energy function is developed in the form of a convolution integral, already used by some research groups (Rashid et al. 2012a; Miller and Chinzei 2002; Prange and Margulies 2002).

Thus, in order to model the brain’s nonlinear elasticity and the time-dependent behaviour, the one-term Ogden hyperelastic model and a Prony-series are combined:

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