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Journal of Applied Mathematics
Volume 2012 (2012), Article ID 516060, 22 pages
Research Article

Fabric Tensor Characterization of Tensor-Valued Directional Data: Solution, Accuracy, and Symmetrization

State Key Laboratory of Hydroscience and Engineering, Tsinghua University, Beijing 100084, China

Received 21 April 2012; Revised 5 July 2012; Accepted 19 July 2012

Academic Editor: Luis Javier Herrera

Copyright © 2012 Kuang-dai Leng and Qiang Yang. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.


Fabric tensor has proved to be an effective tool statistically characterizing directional data in a smooth and frame-indifferent form. Directional data arising from microscopic physics and mechanics can be summed up as tensor-valued orientation distribution functions (ODFs). Two characterizations of the tensor-valued ODFs are proposed, using the asymmetric and symmetric fabric tensors respectively. The later proves to be nonconvergent and less accurate but still an available solution for where fabric tensors are required in full symmetry. Analytic solutions of the two types of fabric tensors characterizing centrosymmetric and anticentrosymmetric tensor-valued ODFs are presented in terms of orthogonal irreducible decompositions in both two- and three-dimensional (2D and 3D) spaces. Accuracy analysis is performed on normally distributed random ODFs to evaluate the approximation quality of the two characterizations, where fabric tensors of higher orders are employed. It is shown that the fitness is dominated by the dispersion degree of the original ODFs rather than the orders of fabric tensors. One application of tensor-valued ODF and fabric tensor in continuum damage mechanics is presented.