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On Obtaining Effective Transversely Isotropic Elasticity Tensors

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Abstract

We consider the problem of finding the transversely isotropic elasticity tensor closest to a given elasticity tensor with respect to the Frobenius norm. A similar problem was considered by other authors and solved analytically assuming a fixed orientation of the natural coordinate system of the transversely isotropic tensor. In this paper we formulate a method for finding the optimal orientation of the coordinate system—the one that produces the shortest distance. The optimization problem reduces to finding the absolute maximum of a homogeneous eighth-degree polynomial on a two-dimensional sphere. This formulation allows us a convenient visualization of local extrema, and enables us to find the closest transversely isotropic tensor numerically.

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Authors and Affiliations

  1. Department of Mathematics and Statistics, Memorial University, St. John’s, NL A1C 5S7, Canada

    Mikhail Kochetov

  2. Department of Earth Sciences, Memorial University, St. John’s, NL A1B 5S7, Canada

    Michael A. Slawinski

Authors
  1. Mikhail Kochetov
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  2. Michael A. Slawinski
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Correspondence to Michael A. Slawinski.

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Kochetov, M., Slawinski, M.A. On Obtaining Effective Transversely Isotropic Elasticity Tensors. J Elasticity 94, 1–13 (2009). https://doi.org/10.1007/s10659-008-9180-2

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  • DOI: https://doi.org/10.1007/s10659-008-9180-2

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