mechmean.orientation_averager
Orientation averager
- class mechmean.orientation_averager.AdvaniTucker(N4)[source]
Orientation averaging following [Advani1987]
The orientation average of a fourth order tensor \(\left<\mathbb{A}\right>_{\text{orientation}}\) is calculated based on fabric tensors of the first kind of order two \(\boldsymbol{N}\) and four \(\mathbb{N}\).
If the symmetry axis of \(\mathbb{A}\) is aligned with the base direction of the first tensor index \(\boldsymbol{e}_{0}\), the average is given by
\[\begin{split}\begin{align} \left<\mathbb{A}\right>_{\text{orientation}} &= b_0\mathbb{N} \\ &+ b_1 \left( \boldsymbol{N} \otimes \boldsymbol{I} + \boldsymbol{I} \otimes \boldsymbol{N} \right) \\ &+ b_2 \left( \boldsymbol{N} \Box \boldsymbol{I} + \left( \boldsymbol{N} \Box \boldsymbol{I} \right)^{T_\text{R}} + \left( \boldsymbol{I} \Box \boldsymbol{N} \right)^{T_\text{H}} + \left( \boldsymbol{I} \Box \boldsymbol{N} \right)^{T_\text{R}} \right) \\ &+ b_3 \boldsymbol{I} \otimes \boldsymbol{I} \\ &+ b_4 \mathbb{I}^{\text{S}} \end{align}\end{split}\]with coefficients
\[\begin{split}\begin{align} b_0 &= A_{0000} + A_{1111} - 2A_{0011} - 4A_{0101} \\ b_1 &= A_{0011} - A_{1122} \\ b_2 &= A_{0101} + \frac{1}{2} \left( A_{1122} - A_{1111} \right) \\ b_3 &= A_{1122} \\ b_4 &= A_{1111} - A_{1122} \end{align}\end{split}\]References
- Advani1987
Advani, S.G. and Tucker III, C.L., 1987. The use of tensors to describe and predict fiber orientation in short fiber composites. Journal of rheology, 31(8), pp.751-784.