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Rate-dependent constitutive modeling of brain tissue

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Abstract

In this paper, the dynamic behavior of bovine brain tissue, measured from a set of in vitro experiments, is investigated and represented through a nonlinear viscoelastic constitutive model. The brain samples were tested by employing unconfined compression tests at three different deformation rates of 10, 100, and 1000 mm/s. The tissue exhibited a significant rate-dependent behavior with different compression speeds. Based on the parallel rheological framework approach, a nonlinear viscoelastic model that captures the key aspects of the rate dependency in large-strain behavior was introduced. The proposed model was numerically calibrated to the tissue test data from three different deformation rates. The determined material parameters provided an excellent constitutive representation of tissue response in comparison with the test results. The obtained material parameters were employed in finite element simulations of tissue under compression loadings and successfully verified by the experimental results, thus demonstrating the computational compatibility of the proposed material model. The results of this paper provide groundwork in developing a characterization framework for large-strain and rate-dependent behavior of brain tissue at moderate to high strain rates which is of the highest importance in biomechanical analysis of the traumatic brain injury.

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Acknowledgements

Special thanks to the Department of Animal Science Department at North Dakota State University for providing the brain tissues. The work was carried out in accordance with the IRB guidelines.

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  1. Department of Mechanical Engineering, North Dakota State University, Fargo, ND, 58104, USA

    Mohammad Hosseini-Farid, Mohammadreza Ramzanpour, Jayse McLean, Mariusz Ziejewski & Ghodrat Karami

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  1. Mohammad Hosseini-Farid
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Appendix

Appendix

1.1 Modeling parallel rheological framework in ABAQUS

Parallel rheological framework (PRF) is a finite-strain constitutive approach, which is referred to model nonlinear viscoelasticity, Mullins effect, and permanent set in polymers and elastomeric materials. The framework is made from the superposition of elastic or elastoplastic networks in parallel with one or multiple finite-strain viscoelastic network, N (Fig. 9). This material model has been available in ABAQUS 6.13 and later versions (ABAQUS/Standard User’s Manual, version 6.13; Providence, RI). This framework can be consisted of arbitrary number of viscoelastic networks and is able to (1) use a hyperelastic material model to specify the elastic response; (2) be combined with Mullins effect to predict material softening; (3) incorporate nonlinear kinematic hardening with multiple back stresses in the elastoplastic response; and (4) employ multiplicative split of the deformation gradient and a flow rule derived from a creep potential to specify the viscous behavior.

Fig. 9
figure 9

Schematic representation of the parallel rheological framework with N viscoelastic networks in parallel with an elastoplastic network

Full size image

A variety of hyperelastic models (e.g., Mooney–Rivlin, neo-Hookean, Ogden, polynomial, Yeoh, etc.) in combination with various common creep laws (i.e., power law, strain-hardening power law, hyperbolic sine, Bergstrom–Boyce) or even with a user-defined creep model led to introduce a diverse material models for representing the complex viscoelastic behavior for materials.

1.2 Assign strain-hardening power-law model in ABAQUS

ABAQUS command Viscoelastic, Nonlinear, LAW = STRAIN was used to define the nonlinear viscoelastic model based on the strain-hardening power-law formulation. The strain-hardening power law is defined by specifying three material parameters: A, n, and m. To obtain physically reasonable behavior, A and n must be positive and −1 < m ≤ 0. In this study, the calibrated material model is implemented as the form of the ABAQUS inp-file format that is shown in the following:

**MATERIALS

**

*Material, name=Ogden-Strain-Law

*Hyperelastic, Ogden

2941., 1.58, 1.37 e−5

**

*Viscoelastic, Nonlinear, NetworkId=1, SRatio=0.727, Law=strain

0.02429, 0.5875, −0.1828

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Hosseini-Farid, M., Ramzanpour, M., McLean, J. et al. Rate-dependent constitutive modeling of brain tissue. Biomech Model Mechanobiol 19, 621–632 (2020). https://doi.org/10.1007/s10237-019-01236-z

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