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