Vol. 16, No. 4, 2021

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The closest isotropic, cubic and transversely isotropic stiffness and compliance tensor to an arbitrary anisotropic material

Xinyuan Shao, Peter D. Folkow and Morteza Eskandari-Ghadi

Vol. 16 (2021), No. 4, 451–470
Abstract

The aim of this paper is to provide, in the framework of Green elasticity, the closest or nearest fourth-order isotropic, cubic and transversely isotropic elasticity tensors with higher symmetries for a general anisotropic elasticity tensor or any other tensors with lower symmetry. Using a gauge parameter, the procedure is done on a dimensionless form based on different generalized Euclidean distances, namely conventional, log-, and power-Euclidean distance functions. In the case of power-Euclidean distance functions, results are presented for powers of 0.5, 1 and 2. Except for the conventional distance function, the different generalized distance functions adopted in this paper preserve the property of invariance by inversion, meaning that the results for the closest stiffness tensor are also valid for the compliance tensor. Explicit formulations are given for determining the closest isotropic and cubic tensors, where the multiplication tables of the bases are diagonal. More involved coupled equations are given for the coefficients of the closest transversely isotropic elasticity tensors, which can be solved numerically. Two different material cases are studied in the numerical examples, which illustrate the material coefficients and error measures based on the present methods, including the influence from the gauge parameter.

Keywords
conventional distance, log-distance, power-Euclidean distance, closest elasticity tensor, linear vector space
Milestones
Received: 14 August 2020
Revised: 7 May 2021
Accepted: 1 June 2021
Published: 9 November 2021
Authors
Xinyuan Shao
Applied Mechanics Masters Program
Chalmers University of Technology
41296 Gothenburg
Sweden
Peter D. Folkow
Department of Mechanics and Maritime Sciences
Chalmers University of Technology
41296 Gothenburg
Sweden
Morteza Eskandari-Ghadi
School of Civil Engineering
University of Tehran
Tehran 1417466191
Iran