This tutorial shows how to reproduce the Holzapfel-Gasser-Ogden material using solid mixtures in FEBio. The main features of this material are as follows:

- Two fiber bundles, making an angle ąPHI with the circumferential direction of the artery. Each fiber follows an exponential function of the fiber stretch, with material coefficients k1 (the initial fiber modulus) and k2 (the exponential coefficient).
- A neo-Hookean ground matrix with a shear modulus MU.
- Incompressibility.

Incompressibility can be simulated by using uncoupled materials in FEBio. Therefore, the neo-Hookean ground matrix can be modeled using the "Mooney-Rivlin" material in FEBio, with c1=MU and c2=0.

The fibers can be modeled using the "fiber-exp-pow-uncoupled" material in FEBio, whose material coefficients relate to the Holzapfel model as ksi=k1, alpha=k2, and beta=2.

Since fibers are defined relative to the circumferential direction, it is necessary to establish a local coordinate system for each element in the mesh using the <mat_axis> tag. In the example of this tutorial (two concentric cylindrical layers representing the media and adventitia of an artery, generated with PreView), the mesh elements have a consistent local node numbering that makes it easy to define the desired local coordinate system. Using <mat_axis type="local">1,4,5</mat_axis>, the local element coordinate system {x1,x2,x3} aligns with the {circumferential, axial, radial} directions of the cylindrical arterial wall.

HolzapfelJBE04.png

In this local cartesian coordinate system {x1,x2,x3}, the fiber directions are specified using spherical angles theta and phi in the definition of a "fiber-exp-pow-uncoupled". The spherical angle phi is the angle between the local axis x3 and the fiber direction. The spherical angle theta is the angle between x1 and the projection of the fiber onto the x1-x2 plane. Therefore, for this model, phi=90 degrees and theta=ąPHI (where PHI is the fiber orientation defined in the Holzapfel model).

The ground matrix and each of two fibers may be combined into a "uncoupled solid mixture" to reproduce the desired features of a Holzapfel-Gasser-Ogden material in FEBio, as shown below:

The material properties for the media layer are taken from Table 1, Expt. no. 71, of Holzapfel, G. A., Gasser, T. C., and Ogden, R. W., 2004. Comparison of a multi-layer structural model for arterial walls with a fung-type model, and issues of material stability. J Biomech Eng 126, 264-275. (The actual experimental data is from Fung, Y. C., Fronek, K., and Patitucci, P., 1979. Pseudoelasticity of arteries and the choice of its mathematical expression. Am J Physiol 237, H620-631).Code:<material id="1" name="Media" type="uncoupled solid mixture"> <mat_axis type="local">1,4,5</mat_axis> <solid type="Mooney-Rivlin"> <density>1</density> <c1>3.831</c1> <c2>0</c2> <k>3831</k> </solid> <solid type="fiber-exp-pow-uncoupled"> <alpha>0.3579</alpha> <beta>2</beta> <ksi>5.3992</ksi> <theta>20</theta> <phi>90</phi> <k>5399.2</k> </solid> <solid type="fiber-exp-pow-uncoupled"> <alpha>0.3579</alpha> <beta>2</beta> <ksi>5.3992</ksi> <theta>-20</theta> <phi>90</phi> <k>5399.2</k> </solid> </material>

Note that the bulk modulus <k> included in each <solid> of the uncoupled solid mixture is taken to be 1000 times c1 and 1000 times k1, to enforce near incompressibility.

The attached sample file, HolzapfelJBE04.feb, provides a complete analysis where the properties of the adventitia layer are also provided, taken from the same paper. In this analysis, the artery is first extended axially to a stretch ratio of 1.6, then inflated with an internal pressure of 30 kPa. Length units are in mm, force in mN, stress in kPa. The results may be compared with Fig. 2 of the 2004 J Biomech Eng paper referenced above, but note that the actual analysis in that paper employed a membrane approximation, whereas the FEBio example included here is a full 3D analysis, therefore results are similar but not identical.

Gerard Ateshian