Evaluation of the Tangent Stiffness Matrix for Hyperelastic Fibres using Automatic Differentiation
Euan Muir  1, *@  , Chris Pearce  1, *@  , Lukasz Kaczmarczyk  2, *@  
1 : University of Glasgow  (Glasgow Univ.)  -  Website
Glasgow G12 8QQ Scotland -  United Kingdom
2 : School of Engineering, The University of Glasgow  (Glasgow Univ.)  -  Website
Glasgow G12 8QQ Scotland -  United Kingdom
* : Corresponding author

Fibres are present in many materials and serve to enhance the material in particular directions. In natural materials, this directional dependency is a result of optimisation whereby the increased strength and stiffness is only where it's needed. Synthetic materials such as fibre reinforced polymers (FRP) have been designed to deliberately exploit this benefit. This paper focuses on fibres in soft tissue but also has broader applicability. The numerical modelling of fibres undergoing finite deformations requires the implementation of a constitutive law and the solution of the governing PDE requires an accurate linearisation of the nonlinear response. In order to ensure a computationally efficient implementation and to simplify the implementation, automatic differentiation has been utilised. 

Automatic differentiation is a process by which the derivatives of a function can be evaluated numerically through the recursive application of the chain rule, exploiting the ability to express the function as basic arithmetic operators and functions. The process itself involves assigning independent and dependent variables and seeding to specify the subject of the differentiation and recording the process for it to be optimised and repeated. 

In this work, the Eberlein fibre model is adopted. This is a hyperelastic material which has two families of fibres and can be used to represent the collagen fibres present in the annulus of the intervertebral disc. The collagen fibres are embedded in a bulk substance which can be represented as a neo-Hookean material [1]. Automatic differentiation will be used to evaluate the tangent stiffness matrix of the Eberlein fibre material in the finite element method.

The open-source package used for the automatic differentiation is ADOL-C [2] and is integrated into our open source finite element software package MoFEM [3].


[1] Eberlein, R., Holzapfel, G., Schulze-Bauer, C. An Anisotropic Model for Annulus Tissue and Enhanced Finite Element Analyses of Intact Lumbar Disc Bodies. In: Computer Methods in Biomechanics and Biomedical Engineering 4.3, pp. 209–229, 2001.

[2] Walther, A., Griewank, A: Getting started with ADOL-C. In U. Naumann und O. Schenk, Combinatorial Scientific Computing, Chapman-Hall CRC Computational Science, pp. 181-202, 2012.

[3] MoFEM, Mesh Orientated Finite Element Method http://bitbucket.org/likask/mofem-joseph

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