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Transversely isotropic higher-order averaged structure tensors

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Abstract

For composites or biological tissues reinforced by statistically oriented fibres, a probability distribution function is often used to describe the orientation of the fibres. The overall effect of the fibres on the material response is accounted for by evaluating averaging integrals over all possible directions in space. The directional average of the structure tensor (tensor product of the unit vector describing the fibre direction by itself) is of high significance. Higher-order averaged structure tensors feature in several models and carry similarly important information. However, their evaluation has a quite high computational cost. This work proposes to introduce mathematical techniques to minimise the computational cost associated with the evaluation of higher-order averaged structure tensors, for the case of a transversely isotropic probability distribution of orientation. A component expression is first introduced, using which a general tensor expression is obtained, in terms of an orthonormal basis in which one of the vectors coincides with the axis of symmetry of transverse isotropy. Then, a higher-order transversely isotropic averaged structure tensor is written in an appropriate basis, constructed starting from the basis of the space of second-order transversely isotropic tensors, which is constituted by the structure tensor and its complement to the identity.

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Author notes

  1. Kotaybah Hashlamoun

    Present address: Graduate Programme in Biomedical Engineering, The University of Calgary, 2500 University Drive NW, Calgary, AB , T2N 1N4, Canada

Authors and Affiliations

  1. Graduate Programme in Mechanical Engineering, The University of Calgary, 2500 University Drive NW, Calgary, AB , T2N 1N4, Canada

    Kotaybah Hashlamoun

  2. Department of Mechanical and Manufacturing Engineering, The University of Calgary, 2500 University Drive NW, Calgary, AB , T2N 1N4, Canada

    Salvatore Federico

Authors
  1. Kotaybah Hashlamoun
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  2. Salvatore Federico
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Correspondence to Salvatore Federico.

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Dedicated to the memory of Prof. Gaetano Giaquinta (Catania, Italy, 25 November 1945–13 August 2016).

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Hashlamoun, K., Federico, S. Transversely isotropic higher-order averaged structure tensors. Z. Angew. Math. Phys. 68, 88 (2017). https://doi.org/10.1007/s00033-017-0830-8

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  • DOI: https://doi.org/10.1007/s00033-017-0830-8

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