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The mechanical behavior of bovine spinal cord white matter under various strain rate conditions: tensile testing and visco-hyperelastic constitutive modeling

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Abstract

The mechanical behavior of the white matter is important for estimating the damage of the spinal cord during accidents. In this study, we conducted uniaxial tension testing in vitro of bovine spinal cord white matter under extremely high strain rate conditions (up to 100 s−1). A visco-hyperelastic constitutive law for modeling the strain rate-dependent behavior of the bovine spinal cord white matter was developed. A set of material constants was obtained using a Levenberg–Marquardt fitting algorithm to match the uniaxial tension experimental data with various strain rates. Our experimental data confirmed that the modulus and tensile strength increased when the strain rate is higher. For the extremely high strain rate condition (100 s−1), we found that both the modulus and failure stress significantly increased compared with the low strain rate case. These new data in terms of mechanical response at high strain rate provide insight into the spine injury mechanism caused by high-speed impact. Moreover, the developed constitutive model will allow researchers to perform more realistic finite element modeling and simulation of spinal cord injury damage under various complicated conditions.

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Data availability

The datasets used or analyzed during the current study are available from the corresponding author on reasonable request.

Abbreviations

\(\lambda\) :

Stretch

\({\varvec{\sigma}}\) :

Cauchy stress tensor

\({{\varvec{\sigma}}}_{{\varvec{h}}}\) :

Cauchy stress tensor of hyperelastic unit

\({{\varvec{\sigma}}}_{{\varvec{v}}}\) :

Cauchy stress tensor of viscoelastic unit

\(W\) :

Strain energy potential function

\({\varvec{F}}\) :

Deformation gradient tensor

\({\varvec{X}}\) :

Initial position

\({\varvec{x}}\) :

Current position

\({u}_{i}\) :

The ith component of the displacement

\({\varvec{C}}\) :

Right Cauchy–Green deformation tensor

\({\varvec{E}}\) :

Green–Lagrangian strain tensor

\({\varvec{I}}\) :

Identity matrix

\({I}_{i}\) :

Invariants of \({\varvec{C}}\)

\({\varvec{S}}\) :

The Second Piola–Kirchhoff stress tensor

\({p}_{h}\) :

Hydrostatic pressure

J :

Determinant of \({\varvec{C}}\)

c ij :

Material constants

\(\boldsymbol{\Omega }\) :

Tensor function describing the strain history effects on stress

\({\sigma }_{ij}\) :

Components of \({\varvec{\sigma}}\)

\(\tau\) :

Integral variable of time

\(U\) :

Kernel function of deformation, time, and deformation rate

\(m(t)\) :

Memory function

\({m}_{i}\) :

Weight factors satisfying \({\sum }_{i=1}^{N}{m}_{i}=1\)

\({\theta }_{i}\) :

Relaxation times

\(\zeta\) :

History-dependent stretch

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Funding

This study was supported by a Grant-in-Aid for Scientific Research (C) from the JSPS (grant 21560107).

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Authors and Affiliations

  1. Department of Mechanical Engineering, Graduate School of Sciences and Technology for Innovation, Yamaguchi University, 2-16-1 Tokiwadai, Ube, Yamaguchi, 755-8611, Japan

    Fei Jiang, Yoshikatsu Onomoto, Junji Ohgi & **an Chen

  2. Department of Mechanical and Electrical Engineering, National Institute of Technology, Tokuyama College, Gakuendai, Shunan, Yamaguchi, 745-8585, Japan

    Itsuo Sakuramoto

  3. Department of Orthopedic Surgery, Yamaguchi University Graduate School of Medicine, 1-1-1, MinamiKogushi, Yamaguchi, 755-8505, Ube City, Japan

    Norihiro Nishida

Authors
  1. Fei Jiang
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  2. Itsuo Sakuramoto
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  4. Yoshikatsu Onomoto
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  5. Junji Ohgi
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Contributions

Fei Jiang wrote and prepared the manuscript. Itsuo Sakuramoto and Junji Ohgi conducted the experiment. Fei Jiang and Yoshikatsu Onomoto developed the constitutive model. Norihiro Nishida and **an Chen gave advice. All authors read and approved the final manuscript.

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Correspondence to Fei Jiang.

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Jiang, F., Sakuramoto, I., Nishida, N. et al. The mechanical behavior of bovine spinal cord white matter under various strain rate conditions: tensile testing and visco-hyperelastic constitutive modeling. Med Biol Eng Comput 61, 1381–1394 (2023). https://doi.org/10.1007/s11517-023-02787-1

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