Fiber reinforced materials.

This is a model which was conceived by G. Holzapfel et al. [37] to model arterial walls. It is an anisotropic hyperelastic model, consisting of an isotropic neo-Hooke potential for the base material, complemented by exponential strenghtening terms in fiber direction. The mathematical form of the potential satisfies:

$$U=C_{10}(\bar{I}_1-3)+\frac{1}{D_1}(J-1)^2+\sum_{i=1}^{n} \frac{k_{1i}}{2k_{2i}} \left[ e^{k_{2i}\langle \bar{J}_{4i}-1 \rangle ^2}-1 \right]$$ (459)

where $\langle x \rangle=0$ for $x < 0$ and $\langle x \rangle=x$ for $x \ge 0$. Thus, the fibers do not take up any force under compression. Although the material was originally defined for arteries, it is expected to work well for other fiber reinforced materials too, such as reinforced nylon. The material model implemented thus far can cope with up to 4 different fibers. The material definition consists of a *MATERIAL card defining the name of the material. This name HAS TO START WITH ”ELASTIC_FIBER” but can be up to 80 characters long. Thus, the last 67 characters can be freely chosen by the user. Within the material definition a *USER MATERIAL card has to be used satisfying:

First line:

• *USER MATERIAL
• Enter the CONSTANTS parameter and its value. The value of this parameter is 2+4n, where n is the number of fiber directions.

Following line if one fiber direction is selected:

• $C_{10}$.
• $D_{1}$.
• $n_{x1}$: x-direction cosine of fiber direction.
• $n_{y1}$: y-direction cosine of fiber direction.
• $k_{11}$.
• $k_{21}$.
• Temperature.

Repeat this line if needed to define complete temperature dependence. The z-direction cosine of the fiber direction is determined from the x- and y-direction cosine since the direction norm is one. If a local axis system is defined for an element consisting of this material (with *ORIENTATION)the direction cosines are defined in the local system.

If more than one fiber direction is selected (up to a maximum of four), the four entries characterizing fiber direction 1 are repeated for the subsequent directions. Per line no more than eight entries are allowed. If more are needed, continue on the next line.

Example:

*MATERIAL,NAME=ELASTIC_FIBER
*USER MATERIAL,CONSTANTS=18
1.92505,0.026,0.,0.7071,2.3632,0.8393,0,-0.7071,
2.3632,0.8393,0.7071,0.,2.3632,0.8393,-0.7071,0.,
2.3632,0.8393

defines an elastic fiber materials with four different fiber directions (0,0.7071,0.7071), (0,-0.7071,0.7071), (0.7071,0.,0.7071) and (-0.7071,0.,0.7071). The constants are $C_{10}=1.92505,~ D_{1}=0.026$ and $k_{1i}=2.3632,~ k_{2i}=0.8393~ \forall~ i~ \in~ \lbrace 1,2,3,4 \rbrace$.