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Vector and Tensor Calculus
Published in Tasos C. Papanastasiou, Georgios C. Georgiou, Andreas N. Alexandrou, ViscousFluid Flow, 2021
Tasos C. Papanastasiou, Georgios C. Georgiou, Andreas N. Alexandrou
The symmetric tensor D=12[∇u+(∇u)T] is the rate of strain (or rate of deformation) tensor, and represents the state of the intensity or rate of strain. The antisymmetric tensor S=12[∇u−(∇u)T] is the vorticity tensor.8 If n is the unit normal to a surface, then the dot product n · D yields the rate of change of the distances in three mutually perpendicular directions. The dot product n · S gives the rate of change of orientation along these directions.
Tensor Algebra
Published in John G. Papastavridis, Tensor Calculus and Analytical Dynamics, 2018
Axial vector of an (absolute) antisymmetric tensor. In three dimensions, such a tensor has, at most, three independent components: in covariant form (and similarly for contravariant and/or mixed forms): T=(Tkl=−Tlk)=(0T12T13−T120T23−T13−T230)
Fourth-rank Tensors: Elasticity
Published in D. R. Lovett, Tensor Properties of Crystals, 2018
Rewrite the following tensor as the sum of a symmetric and an antisymmetric tensor: [188562411127].
Surface anchoring energy of cholesteric liquid crystals
Published in Liquid Crystals, 2020
Tianyi Guo, Xiaoyu Zheng, Peter Palffy-Muhoray
where is the Levi-Civita antisymmetric tensor. The energy is a proper scalar, allowed by chirality and the presence of the pseduoscalar . Explicitly, in terms of the eigenvectors, it becomes