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Research Papers

Numerical Approximation of Elasticity Tensor Associated With Green-Naghdi Rate

[+] Author and Article Information
Haofei Liu

Department of Mechanics,
Tianjin University,
92 Weijin Road,
Tianjin 300072, China

Wei Sun

Tissue Mechanics Laboratory,
The Wallace H. Coulter Department of Biomedical Engineering,
Georgia Institute of Technology,
Technology Enterprise Park,
Room 206, 387 Technology Circle,
Atlanta, GA 30313-2412
e-mail: wei.sun@bme.gatech.edu

Manuscript received July 26, 2016; final manuscript received May 3, 2017; published online June 16, 2017. Assoc. Editor: Hai-Chao Han.

J Biomech Eng 139(8), 081007 (Jun 16, 2017) (8 pages) Paper No: BIO-16-1317; doi: 10.1115/1.4036829 History: Received July 26, 2016; Revised May 03, 2017

Objective stress rates are often used in commercial finite element (FE) programs. However, deriving a consistent tangent modulus tensor (also known as elasticity tensor or material Jacobian) associated with the objective stress rates is challenging when complex material models are utilized. In this paper, an approximation method for the tangent modulus tensor associated with the Green-Naghdi rate of the Kirchhoff stress is employed to simplify the evaluation process. The effectiveness of the approach is demonstrated through the implementation of two user-defined fiber-reinforced hyperelastic material models. Comparisons between the approximation method and the closed-form analytical method demonstrate that the former can simplify the material Jacobian evaluation with satisfactory accuracy while retaining its computational efficiency. Moreover, since the approximation method is independent of material models, it can facilitate the implementation of complex material models in FE analysis using shell/membrane elements in abaqus.

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Grahic Jump Location
Fig. 1

FE meshes used in the biaxial test simulation: the arrows denote the loading directions with their round roots showing the sites of loading, the box enclosed region in the center represents the area delineated by the four markers in the biaxial test (a), Von Mises stress contour in the deformed configuration (b), and curves of Cauchy stress versus Green strain resulted from experiment (dashed line) and numerical simulation (solid line) (c)

Grahic Jump Location
Fig. 2

Relative error of the Jacobian approximation at different logarithmic perturbations (a), the CPU time (b), and the iteration number (c) at different logarithmic perturbations using themodified HGO model in the biaxial testing simulation. Dashed lines (—) represent the level achieved using UHYPER subroutine.

Grahic Jump Location
Fig. 3

FE meshes used in the vessel inflation simulation, before (a) and after (b) deformation; loading pressure versus the outermost radius of the deformed vessel, observed in experiment (dotted line) and numerical simulations. Material models with various mean fiber directions while the other parameters are kept the same as baseline values (c); material models with various standard deviations while the other parameters are kept the same as baseline values (d); and the iteration number at different logarithmic perturbations using the distributed fiber model in the rat carotid artery simulation (e).

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