Journal
MATHEMATICS AND MECHANICS OF SOLIDS
Volume 24, Issue 10, Pages 3092-3102Publisher
SAGE PUBLICATIONS LTD
DOI: 10.1177/1081286519835246
Keywords
Invariant; irreducible functional basis; symmetric and traceless tensor; fourth-order tensor
Categories
Funding
- National Natural Science Foundation of China [11701132, 11571178, 11771405, 11372124]
- Hong Kong Research Grant Council [PolyU 15302114, 15300715, 15301716, 15300717]
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The elasticity tensor is one of the most important fourth-order tensors in mechanics. Fourth-order three-dimensional symmetric and traceless tensors play a crucial role in the study of the elasticity tensor. In this paper, we present two isotropic irreducible functional bases for a fourth-order three-dimensional symmetric and traceless tensor. One of them is exactly the minimal integrity basis introduced by Smith and Bao in 1997. It has nine homogeneous polynomial invariants of degrees two, three, four, five, six, seven, eight, nine and ten, respectively. We prove that it is also an irreducible functional basis. The second irreducible functional basis also has nine homogeneous polynomial invariants. It has no quartic invariant but has two sextic invariants. The other seven invariants are the same as those of the Smith-Bao basis. Hence, the second irreducible functional basis is not contained in any minimal integrity basis.
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