A two-layer elasto-visco-plastic rheological model for the material parameter identification of bone tissue

Research output: Journal article (peer-reviewed)Journal article

11 Citations (Scopus)


The ability to measure bone tissue material properties plays a major role in diagnosis of diseases and material modeling. Bone's response to loading is complex and shows a viscous contribution to stiffness, yield and failure. It is also ductile and damaging and exhibits plastic hardening until failure. When performing mechanical tests on bone tissue, these constitutive effects are difficult to quantify, as only their combination is visible in resulting stress-strain data. In this study, a methodology for the identification of stiffness, damping, yield stress and hardening coefficients of bone from a single cyclic tensile test is proposed. The method is based on a two-layer elasto-visco-plastic rheological model that is capable of reproducing the specimens' pre- and postyield response. The model's structure enables for capturing the viscously induced increase in stiffness, yield, and ultimate stress and for a direct computation of the loss tangent. Material parameters are obtained in an inverse approach by optimizing the model response to fit the experimental data. The proposed approach is demonstrated by identifying material properties of individual bone trabeculae that were tested under wet conditions. The mechanical tests were conducted according to an already published methodology for tensile experiments on single trabeculae. As a result, long-term and instantaneous Young's moduli were obtained, which were on average 3.64 GPa and 5.61 GPa, respectively. The found yield stress of 16.89 MPa was lower than previous studies suggest, while the loss tangent of 0.04 is in good agreement. In general, the two-layer model was able to reproduce the cyclic mechanical test data of single trabeculae with an root-mean-square error of 2.91 ± 1.77 MPa. The results show that inverse rheological modeling can be of great advantage when multiple constitutive contributions shall be quantified based on a single mechanical measurement.

Original languageEnglish
Pages (from-to)2149-2162
Number of pages14
JournalBiomechanics and Modeling in Mechanobiology
Issue number6
Publication statusPublished - 01 Dec 2020


  • Algorithms
  • Biomechanical Phenomena
  • Bone and Bones/physiopathology
  • Elastic Modulus
  • Elasticity
  • Female
  • Femur/pathology
  • Humans
  • Materials Testing
  • Middle Aged
  • Plastics
  • Rheology
  • Stress, Mechanical
  • Tensile Strength
  • Viscosity


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